Hitler’s Math

This is the first article in an occasional series, Premonitions of Disaster, on the lives of German Jews in the 1920s and 1930s. Abraham Fraenkel was born in Germany in 1891, and died in Israel in 1965, aged 74. His Recollections of a Jewish Mathematician in Germany were reissued last year by Springer.

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My report about this last phase of my life in Germany should not close without my describing some people who in every respect deserve to be highlighted. Those who first come to mind are eight scientists. Of course, I cannot and do not wish to offer biographies or acknowledgments of their scientific accomplishments that can be easily found elsewhere. Instead, I will mention primarily those aspects that were significant for my own development. Of these eight men, there are four mathematicians: Hilbert, Brouwer, Landau, and von Neumann; two physicists: Einstein and Niels Bohr; and two Protestant theologians and philosophers: Rudolf Otto and Heinrich Scholz.

In his time David Hilbert (1862–1943) was the most significant mathematician in the world. For a long time, he shared this honor with Henri Poincaré, who died in 1912. In contrast to most of his colleagues, Hilbert’s discoveries in successive periods encompassed the broadest range of pure mathematics. He hardly dealt with applied mathematics, except for one not very successful period devoted to physics. He was born in Königsberg and never relinquished his East Prussian accent. The number of true anecdotes about him is legion, as he was, without doubt, a highly original character. He became a professor in Göttingen in 1895 and declined appointments to Leipzig, Berlin, Heidelberg, and in 1919 to Bern. He was correctly considered the scientific head of German mathematics, and was acknowledged throughout the world. Students flocked to him from all over Europe and the United States. At the second International Congress of Mathematicians in Paris in 1900 he gave a programmatic lecture on “Mathematical Problems.” The 23 key unsolved problems he enumerated largely determined the developments in mathematics in the subsequent decades. Most of these problems have since been solved, problem No. 1 by Paul J. Cohen in 1963.

Hilbert always remained free of all national and racist prejudices. After the turn of the 20th century, he had a large number of Jewish students, both in absolute and relative terms. In his own working life, he was greatly influenced by two Jews, Adolf Hurwitz and Hermann Minkowski. His authority and tenacity managed twice to break through the prewar prejudice in the Prussian Ministry of Education and Cultural Affairs against appointing Jews to full professorships: Minkowski in 1902 and Landau in 1909. He pushed through the postdoctoral professorial qualification (Habilitation) granted Emmy Noether at the University of Göttingen despite the resistance of many colleagues, including some Jews, although this resistance was chiefly directed to her being a woman, rather than on account of her being a Jew and salon communist. Hilbert’ s response to a question of Bernhard Rust, the Nazi Reich Minister for Science, Education, and Popular Culture, was typical. At a banquet in 1934 in Göttingen, Rust asked: “Is it really true, Mr. Professor, that your institute suffered so much from the departure of the Jews and their friends?” to which Hilbert replied, in his characteristic East Prussian dialect: “Suffered? No, it hasn’t suffered, Mr. Minister. It simply doesn’t exist anymore!”

I never studied in Göttingen so I was never actually a student of Hilbert. It is thanks to a fortuitous occurrence that I nevertheless had the opportunity to develop a close relationship with him. His last creative period, starting in 1918, was dedicated to axiomatics and the foundation of mathematics in general. Totally unrelated to that, in 1921, I had taken on the axiomatics of set theory, the fundamental discipline of mathematics, to which Hilbert had always attached great significance, and whose creator, Georg Cantor, he had defended against harsh criticism. Hilbert’ s highly talented assistant and close collaborator Paul Bernays (descendant of the famous Hamburg Talmud scholar Chacham [Isaac] Bernays)—who taught at the Swiss Institute of Technology (ETH) in Zurich starting in 1933—focused as of 1935 specifically on set theory. As a result of my own engagement with set theory, Hilbert asked me in 1922 to give lectures at the Göttingen Mathematical Society. A relentless and yet objective critic on such occasions, he must have been impressed with my results, because he offered me warm praise. Our further communication took place mostly in writing. Although he had accepted my invitation to hold a lecture in Kiel, due to external circumstances this did not come to pass. After the third edition of my Einleitung in die Mengenlehre (Foundations of Set Theory) was published, I received a very cordial handwritten letter of thanks. Not until much later did I learn that he had considerable influence on my receiving the appointment in 1928 as a full professor in Kiel, since the Ministry had sought his opinion on the various candidates.

L.E.J. Brouwer [1881–1966] of Holland can be viewed as the decided adversary of Hilbert. His mathematical merits are limited to two—to be sure, very important— achievements. He mastered, especially from 1911 to 1914, certain difficult topographical problems that have to do with the concept of dimension; however, this was the virtual extent of his actual mathematical work. But even earlier, and also in the years 1918–1930, he had developed an original doctrine in an ultradogmatic manner that he referred to as “intuitionism,” and later “neointuitionism,” in which he rejected the greatest part of “classical” mathematics of the last three centuries as well as “classical” (Aristotelian) logic, with the argument that these were in part meaningless and in part incorrect teachings, since they were based in particular on “existence” rather than “construction.”

His standpoint was rejected by the overwhelming majority of mathematicians and logicians. His discussions with Hilbert, in particular, assumed a personal note. The two virtually drowned one another with their cursing. The dispute took on a particular nuance in the sense that the Dutchman Brouwer set himself up as a champion of Aryan Germanness. Consequently, Hilbert removed him from the editorial board of the Mathematische Annalen after he objected to what he felt were too many Eastern European Jewish (“Ostjuden”) authors. In 1922 he published a 16-page Open Letter of protest to the Dutch Minister of Education for allowing the outstanding French scholar Arnaud Denjoy, who was professor at the University of Utrecht and a member of the Amsterdam Academy of Sciences, to maintain a “corresponding membership” although Denjoy conducted himself, according to Brouwer, like an “agent for the French government.” Brouwer also openly protested against participating in the International Congress of Mathematicians in Bologna in 1928, claiming that the many German scholars, who under Hilbert’s leadership participated in the Congress along with the French, had sinned against the “Manes of Gauss and Riemann.” Luckily, his instinct saved him—in contrast, mutatis mutandis, to another outstanding Dutch mathematician—from accepting a position at the University of Berlin, which the Nazis offered him in 1933.

I often got together with Brouwer, especially in Amsterdam, and was among the first mathematicians outside of Holland who treated his intuitionist views in lectures, articles, and books in a non-dogmatic, often critical form. At first, Brouwer was enthusiastic in his praise of me, but I later became the object of his passionate and grievous rejection. My presentations of intuitionism were the earliest, but by no means the most in-depth. Starting in 1930 many esteemed mathematicians from Holland, the United States, and other countries distilled from Brouwer’s ideas all that made sense and lacked dogmatic prejudice, and was insightful and sometimes groundbreaking also in classical mathematics.

Brouwer was temporarily suspended in 1945 by the Dutch government because of his connections to the Nazis, especially during the German occupation of Holland, when the Dutch Jews suffered extremely cruel treatment and most were exterminated in camps, but he later regained his position. I found it impossible to resume our former friendly relations.

Edmund Landau (1877–1938) can also be described as an original sort of character. Starting in the 1920s he took on the middle name “Yechezkel,” after the Prague rabbi and Talmudist Yechezkel Landau, whose brother was an ancestor of his. Edmund Landau’ s father was the well-known Berlin gynecologist Professor Leopold Landau, who, after originally being rather distant to Judaism, later espoused the national-Jewish stance and supported the cause of the Palestine/land of Israel Jewish Colonization Association (PICA). He proudly showed me Hebrew inscriptions in the courtyard of his mansion near Berlin’s Pariser Platz.

Landau was one of the most prolific scientists of his time. From 1904 to 1909 alone he published more than 50 papers, some of which were very extensive, and a two-volume, innovative book on the distribution of prime numbers. His main areas of research were the theory of functions and analytical number theory. He worked together with the outstanding English mathematicians G.H. Hardy and John Edensor Littlewood, with Harald Bohr, the brother of Niels Bohr, and many others. Over time he developed an increasingly concise, concentrated style of writing, not only in his papers but also in his books, totally erasing the intuitive dimension of his mathematical problem-solving—in a tradition first introduced by Gauss. Not everyone likes this style, especially when dealing with such difficult subjects. Apart from that, he did not lack a sense of humor, as demonstrated by his prefaces to Foundations of Analysis, which was widely read in both German and English.

In 1899 Landau completed his doctorate and in 1901, at 24, his postdoctoral professorial qualification, both at the University of Berlin. After the premature death of Minkowski, Hilbert wanted a first-class successor, from whom he anticipated creative talents. And so it happened that in 1909, Landau, at 32 years of age, received the important Göttingen professorial position, in every sense an unusual appointment, skipping the then usual intermediate step of nontenured associate professor (Extraordinarius). He was very successful as a teacher in Göttingen and attracted doctoral candidates from all over the world. His relationship to Hilbert, however, turned out not to be as agreeable as had been expected. After Felix Klein (dubbed the “foreign minister of mathematics”) retired, Hilbert appointed the Jewish mathematical analyst Richard Courant as his successor; Courant is credited especially for the establishment of Göttingen’s prestigious Mathematical Institute after the First World War, with funding from the United States. In 1933 special efforts were made to try to keep him, but he was lucky to soon receive an academically and administratively more outstanding position at New York University.

Landau’s avid wish to receive a professorship in Berlin did not come true. His quirky nature had made him many enemies, especially Ludwig Bieberbach, who had enjoyed a full professorship in Berlin since 1921. By virtue of his parents’ and his brother-in-law Paul Ehrlich’ s assets, Landau was a wealthy man. He built a mansion-like home in Göttingen to house his huge scientific library. As a guest, I often had the chance to admire his lavishly designed home. Aside from mathematical research, to which Landau was dedicated with untiring diligence, he had two hobbies: He owned an extensive and probably valuable postage-stamp collection, which he constantly expanded, and in his free time he read detective stories with a passion.

After the founding of the Hebrew University in Jerusalem in 1925 with initially three institutes, it seemed natural that a chair should also be established for mathematics, a science often characterized as “Jewish.” In addition to the two world-famous mathematicians Jacques Hadamard and Edmund Landau, the university’ s Board of Governors also included Selig Brodetsky who in 1924 had taken up a position as professor of applied mathematics at the University of Leeds and who as a Zionist leader had rendered great service to the university. At first Einstein was also a member, but he soon withdrew from the Board of Governors. Shortly after the 14th Zionist Congress in Vienna, the Board of Governors of the Hebrew University met in September 1925 in the home of Dr. Elias Straus in Munich. On the agenda was also the question of the appointment of a mathematics chair. As I remember, it was the last (or even the only) session in which Einstein participated. I was also in Munich at the time, at my father’s deathbed, and Landau visited me. Obviously, there was a proposal that Landau go to Jerusalem either temporarily or long-term. He was in fact offered the position and was prepared to accept it. Although he had not been raised close to Jewish tradition, much less Hebrew scripture, he made an effort in his usual energetic fashion to learn Hebrew and managed amazingly quickly to master the language, especially to speak it fluently. And so he went to Jerusalem in the fall of 1927 as the first professor of mathematics. Supported by associate professor Dr. Benjamin Amira, who was raised in Palestine/Land of Israel and had completed his doctorate at the University of Geneva, Landau successfully taught the eager students for the duration of the semester.

This is not the place to discuss the differences that arose between Landau, the university management, and the Board of Governors. In any case, Landau left Jerusalem at the end of the winter semester and never returned. Thus the Board, when it convened again in June 1928 in London, had once again to place the appointment of the mathematics chair on the agenda.

Edmund Landau had also paid attention to my prewar publications and followed my work from the early 1920s even more closely, although it was in a field that was relatively unrelated to his. In 1927, when I was still a non-tenured associate professor (Extraordinarius) in Marburg, he had already spoken with me somewhat tentatively of his intention to suggest me to the Board of Governors for the professorial chair in Jerusalem. I did not take him too seriously, especially when I heard that Hadamard wanted to suggest the distinguished Polish-French analyst Szolem Mandelbrojt; besides that, Landau’ s own decisions before he left for Jerusalem were anything but firm. Thus it was a great surprise for me when on June 5, 1928, in Kiel, only a few weeks after I assumed my position there, I received a telegram from London offering me the chair in Jerusalem. As Landau later explained to me, Mandelbrojt had decided not to leave France, so his candidacy had never even been considered. And he did, in fact, receive a position in Paris a short time later, as Hadamard’s successor at the Collège de France. My appointment had been unanimous, with even the approval of the physicist on the Board of Governors, Leonard S. Ornstein from the Netherlands.

Landau’s and my relationship became increasingly close in those years, encouraged by his growing competence in Hebrew. He even sent me telegrams in Hebrew from Göttingen to Marburg! My last encounter with Landau was in Switzerland shortly after the Nazis came to power. In contrast to Courant, he was not frightened, but rather filled with an incomprehensible optimism. But the blow came soon— even sooner for him than for other Jewish professors. The Göttingen students demonstrated against him—evidently having been incited from higher up—and demanded his dismissal. A certain lecture sheds light on the background to this “resolute rejection” of Landau. It was held by Berlin professor Ludwig Bieberbach in spring 1934 in Berlin at the annual conference of the Verein zur Förderung des mathematischen und naturwissenschaftlichen Unterrichts. The press, for example, the Deutsche Zukunft (German Future) of April 8, 1934, and Forschungen und Fortschritte (Research and Progress) of June 20, 1934, gave extensive coverage of the lecture. Cited here are some passages from it:

The conduct of the Göttingen students toward Edmund Landau is well-founded and justified, as the Landau case clearly shows that there is a German and a Jewish mathematics, two worlds, separated by an unbridgeable chasm. The selection of the problems and the style of treating them is characteristic of the thinker and therefore a product of his racial affiliation. … A people that has come into its own cannot tolerate such teachers and must reject foreign thought.

Bieberbach then made reference to the psychological typology of Erich Jaensch, assigning the abstracting S type to the Jews and speakers of Romance languages, and that of the J type, which seeks to understand reality in all its manifold aspects, to the Germans. He remarked on the explication of imaginary numbers by Cauchy and Goursat (both of whom are not Jews!):

The Cauchy-Goursat theorem arouses intolerable displeasure in us Germans. But certainly not in a Landau. Such a thing is mentally determined. … All in all, such juggling with concepts and a pronounced shrewdness characterize the nonorganic, hostile S type, especially Jewish mathematics.

This was then compared to the “Eastphalian” Gauss, the “Nordic-Dinaric” Felix Klein, and the “Eastern Baltic–Nordic” Hilbert:

Abstract Jewish thinkers of the S type knew how to deform [the axiomatics of Hilbert], so that they can be used as intellectual varieté. … This is a typical example of how race-alien influences and race-alien seduction blocks the Germans from the source of their own strength. … There is no self-sufficient mathematical domain that is independent of ideology and life; the dispute on the fundamental principles that is now raging is in reality a dispute over race.

The journal Deutsche Zukunft drew its own conclusions:

For practical cultural policy it turns out that mathematics is freed from the curse of sterile intellectualism; its burden falls upon those thinkers who are alien to the nation and the race, who will no longer exist in the future and whose representatives belong to the past and can no longer be viewed as German scientists. German mathematics is rooted in blood and soil.…

These statements caused incredible outrage abroad, perhaps more than did the brutal dismissal of Landau himself and all the Jewish and “half-Jewish” scholars in Germany. In a letter to Bieberbach of May 19, 1934, one of the most eminent American mathematicians, Professor Oswald Veblen of the Institute for Advanced Study in Princeton, New Jersey, expressed his disgust at the distinctions made between Jewish and German mathematics and at such pseudo-scientific justification of the measures taken against Landau, “one of the great mathematicians of our epoch.” He added that the article in Deutsche Zukunft was received in the United States “with varying degrees of sorrow, derision, and contempt.” Veblen had written this letter in an outburst of pure humanitarianism, without having been able at the time to foresee that the expulsion of all “non-Aryan” scholars from Germany, and later also from Austria, would make the hospitable United States (together with Russia) into the center of the world for mathematics and physics, even before the end of the 1930s, particularly in view of modern nuclear physics.

The consequences of this shift are generally known. After losing his professorial chair in Göttingen, Landau moved to Berlin. Although he also went abroad to give lectures, he never managed to decide to emigrate, which would have meant losing all his assets. He died in 1938 at the age of only 61.

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Reprinted from Recollections of a Jewish Mathematician in Germany, by Abraham A. Fraenkel, edited by Jiska Cohen-Mansfield, translated by Allison Brown. Copyright © 2016 Springer International Publishing AG. All rights reserved.