Years Ago

David E. Rowe, Editor

‘‘Mathematics Knows No Races’’: A Political Speech that David Hilbert Planned for the ICM in Bologna in 1928 REINHARD SIEGMUND-SCHULTZE

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, e-mail: [email protected] 1

decade after World War I, the International Congress of Mathematicians (ICM) in Bologna in 1928 finally put an end to the so-called ‘‘boycott of German mathematics,’’ that is, the exclusion of mathematicians coming from Germany and its War allies from official, organized international mathematical collaboration. In the eyes of most mathematicians in all countries, this normalization was long overdue. More than 70 German mathematicians arrived in Bologna in September 1928, which was the largest national contingent of mathematicians second only to the host nation Italy. The unofficial leader of the Germans was David Hilbert (1862-1943), at that time probably the most famous mathematician in the world. From a three-page undated draft (probably handwritten by Hilbert’s wife Ka¨the), now in the Hilbert papers at the Manuscript Division of the State and University Library in Go¨ttingen (Fig. 1),1 one can conclude that in addition to his invited plenary lecture, Hilbert intended to present in Bologna a short political speech, which culminated in the following words: ‘‘Mathematics knows no races [Die Mathematik kennt keine Rassen.]… For mathematics, the whole cultural world is a single country.’’ Strikingly, ‘‘Mathematics knows no races’’ seems to go beyond the national conflicts of the time. Hilbert, who was not Jewish himself, seems farsighted and even seems to foresee and to warn against the Nazi rule five years later, and against Ludwig Bieberbach’s racist theories of ‘‘Deutsche Mathematik’’ after 1933 (Mehrtens 1987). We will later see that this immediate conclusion is somewhat premature. But why do I say ‘‘intended’’? Didn’t Hilbert actually give this political talk? In fact, we have no documentary evidence proving that Hilbert actually addressed the mathematicians in Bologna with these words. To the contrary rather, there is no trace of Hilbert’s political address in the published Italian Proceedings of the Congress (Atti 1929), in any published contemporary report about the event,2 or in any correspondence of contemporaries that has been available to me. Even if the editors of the Proceedings of the Bologna ICM had hesitated, for political or diplomatic reasons, to print the text of Hilbert’s address in detail,3 they surely would have mentioned his effort in some way if it was part of an official event during the Congress. Even if it was an

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Cod. Ms. D. Hilbert 494: 19.

2

See for example (Anon 1928, Blaschke 1930, Bortolotti 1928, Buhl 1928, Fehr 1929, Gingrich 1928, Tonelli 1929). I thank Ulf Hashagen (Munich) and Annalisa Capristo (Rome) for having alerted me to some of these reports. I could not find reports in Jahresbericht DMV and in Science. The most convincing evidence that Hilbert did not give his political speech is that it is not mentioned in Ha¨rlen’s letter to Brouwer, quoted below. Brouwer would most certainly have been very interested to hear about it. There is a slight uncertainty even in this case because in the surviving long letter by Ha¨rlen there is something missing at the end. 3

Such hesitation seems, however, unlikely, given that the Proceedings go otherwise into the details of the nationalistic conflicts before the Congress.

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Figure 1. Excerpt from Hilbert’s manuscript in Ka¨the Hilbert’s hand, Cod. Ms. D. Hilbert 494: 19. (Courtesy Handschriftenabteilung Niedersa¨chsische Staats und Universita¨tsbibliothek Go¨ttingen)

unofficial presentation, for instance during a dinner party or a reception, the address with its central statement ‘‘Mathematics knows no races’’ would likely have been noted in reports or in correspondence at the time, in view of its unusual outspokenness.4 Does it really matter whether Hilbert presented the talk or not, since we know from the manuscript that he intended to do it? It does, because Hilbert was so influential and he would have set an encouraging example. People have all kinds of plans and feelings but keep silent in the end. In Hilbert’s case, however, one does not need to assume that he flinched because of fear to speak out. As I describe later, I believe that Hilbert’s poor health at the time of the Bologna Congress together with certain imbalanced passages in the draft provide an explanation as to why he probably felt unable to present it. I have reason to believe that the organizers (judging from the tone of the Proceedings) would have strongly welcomed Hilbert’s somewhat irregular and unusual presentation and would have gone out of their way to provide him with opportunities to deliver it. In this article I will discuss the content of the manuscript and how its content reflects the political situation of world mathematics as Hilbert saw it at the time. I begin with the 3-page manuscript both in its German original and in the English translation. I reproduce the German orthography of the time and insert brackets for presumed typos in the German text, which is stylistically and grammatically not perfect. I add a few minor explanatory footnotes to the English translation and subsequently provide a more complete historical interpretation.

The Text of Hilbert’s Intended Speech in its German Original5 ‘‘M[eine] H[erren]. Ich freue mich dass hier seit langer schwerer Zeit alle Mathematiker der Welt vertreten sind wie es sich geho¨rt und wie es zum Gedeihen unsrer geliebten Wissenschaft no¨thig ist. Bedenken wir dass wir als Mathematiker auf der ho¨chsten Stufe und ho¨chsten Ho¨he der Kultur des strengen Wissens stehen. Wir haben keine Wahl uns anders zu stellen als auf diesen ho¨chsten 4

Standpunkt denn alle Schranken, insbesondere nationale, widerstreben aufs Aeusserste dem Wesen der Mathematik. Es ist ein vollkommnes Missversta¨ndniss Unterschiede, oder gar Gegensa¨tze nach Vo¨lkern oder Menschenrassen zu construiren, die Gru¨nde mit denen man das versucht hat sind sehr fadenscheinig. Die Mathematik kennt keine Rassen. Wenn wir, auch nur oberfla¨chlich auf die Geschichte unserer Wissenschaft schauen so sind alle Nationen und Vo¨lker, die grossen wie die kleinen, gut und gleich darin betheiligt. Denken wir an: Descartes, Fermat, Pascal, Huygens, Newton, Leibnitz, Bernoulli, Euler, d’Alembert, Lagrange, Monge, Laplace, Legendre, Fourier, Gauss, Poisson, Moebius, Chales [sic], Lame´, Steiner, Abel, Jacobi, Dirichlet, Hamilton, Riemann, Clebsch, Cantor, Poincare´, Darboux, Klein—[pageturn] diese Namen sind durcheinandergewu¨rfelt zwischen den Nationen, wie es der Wu¨rfelbecher nicht gru¨ndlicher und unparteiischer ha¨tte thun ko¨nnen. Und wie steht es nun wenn wir von den Gegensta¨nden ausgehen und einzelne Theilgebiete des gewaltigen Reiches der Mathematik herausgreifen? z.B. wo ist Geometrie getrieben worden? Wir kennen Alle die grosse, langandauernde Blu¨tezeit der Geometrie in Frankreich. Wie dann deutsche Mathematiker eingriffen und dann insbesondere die algebraische Geometrie, dieser vielleicht tiefst gelegne Theil der Geometrie in Italien die nachhaltigste Pflege und erfolgreichste Behandlung bis auf den heutigen Tag gefunden hat. Oder Zahlentheorie? fu¨r die uns das urwu¨chsige Russland Tschebischew schenkte, und die Zusammenarbeit von Deutschland und Frankreich—denken wir nur an Jacobi und den grossen, einzigen Hermite—weltbekannt ist. Und dann erst Funktionentheorie! wo Weierstrass die Fundamente erarbeitete. Dann Poincare´, der gla¨nzendste Vertreter der Mathematik wa¨hrend einer ganzen Periode durch seine funktionentheoretischen Entdeckungen die gro¨sste Bewunderung der Welt hervorrief, und derselbe Poincare´ fand seine fleissigsten, enthusiasmirtisten [sic], eifrigsten Schu¨ler in Deutschland. Und wie sehr sind dann wieder die funktionentheoretischen und grossen analytischen Fragen in Italien—[pageturn] behandelt und mit den geometrischen Interessen der italienischen Mathematiker zu einer Einheit verschmolzen worden. Fu¨r die Mathematik ist die gesamte Kulturwelt ein einziges Land. Das wahre Wohl der Wissenschaft vor Augen werden wir nicht u¨ber Zwinsfa¨den [sic for ‘‘Zwirnsfa¨den’’] stolpern und uns nicht durch einzelne Disentirende [sic] beirren lassen, denn so dient auch Jeder seinem Lande am Besten, jedenfalls ist die Liebe zur Mathematik allen Kulturvo¨lkern gemeinsam und besonderer Dank gebu¨rt [sic] den Italienern, die diesen Kongress zu Stande gebracht haben.’’

English Translation (by R. S.-S.) ‘‘G[entlemen]. It makes me very happy that after a long, hard time all mathematicians of the world are represented here. This is as it should be and as it must be for the

There is a slight possibility that Hilbert uttered these words on the eve of the Congress during a reception of the Italian Mathematical Society (U.M.I.) on Sunday, 2 September 1928. But given that Bortolotti in the Bollettino of that society does not mention anything in this respect, this seems very unlikely, too (Bortolotti 1928). 5 The following is a transcription and a translation of the handwritten manuscript, Cod. Ms. D. Hilbert 494: 19. Courtesy Manuscript Division, State-and-University Library Go¨ttingen. The handwriting is apparently by Hilbert’s wife Ka¨the Hilbert.

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prosperity of our beloved science. Let us consider that we as mathematicians stand on the highest pinnacle and highest height6 of the culture of rigorous knowledge.7 We have no other choice than to assume this highest place, because all limits, especially national ones, are contrary to the nature of mathematics. It is a complete misunderstanding [of our science]8 to construct differences or even incompatibilities9 according to peoples and races, and the reasons10 for which this has been done are very shabby ones. Mathematics knows no races. If we look—even superficially—at the history of our science, we see all nations and peoples, the big as well as the small, taking successful and equal part in it. Let us think of Descartes, Fermat, Pascal, Huygens, Newton, Leibniz, Bernoulli, Euler, d’Alembert, Lagrange, Monge, Laplace, Legendre, Fourier, Gauss, Poisson, Moebius, Chasles, Lame´, Steiner, Abel, Jacobi, Dirichlet, Hamilton, Riemann, Clebsch, Cantor, Poincare´, Darboux, Klein—[pageturn] these names are thrown wildly among the nations, as a dice-cup couldn’t do more thoroughly and less biased. And how about the themes themselves, if we select single areas from the monumental kingdom of mathematics? For instance, where has geometry been cultivated? We all know the great, longlasting flourishing of geometry in France. How then German mathematicians got involved and how in particular algebraic geometry—that perhaps deepest part of geometry—found the most lasting cultivation and most successful treatment in Italy until today. Or number theory, for which elemental11 Russia gave us Chebyshev, the area where collaboration between Germany and France—let us just think of Jacobi and the great, unique Hermite—is known worldwide.12 And then, even more striking, [there is] function theory, where Weierstrass laid the foundations. Then Poincare´, the most illustrious representative of mathematics in an entire period, who gained the world’s greatest admiration through his discoveries in function theory; the same Poincare´ found his most diligent, most enthusiastic and most assiduous students in Germany. And again, how much are the function-theoretic and the big analytical questions treated in Italy and forged to a unity with the geometric interests of Italian mathematicians.

For mathematics, the whole cultural world is a single country. Having in mind the true benefit of science we won’t stumble over threads of twine13 and won’t be irritated by isolated dissenters, because this way one also serves best the interests of one’s own country. Anyhow all civilized nations share the love for mathematics and we have to be particularly grateful to the Italians who managed to put together14 this Congress.’’

Ending the Boycott Some nationalist mathematicians on both sides of the political divide opposed the readmission of German mathematicians to the ICM in 1928. The boycott of German and Austrian science after World War I had been spearheaded by the International Research Council (IRC), the science organization of the Entente, whose president from 1919 to 1936 was the French mathematician E´mile Picard (1856-1941). The IRC began admitting members from former enemy nations in 1926. Picard insisted on the connection of the International Congresses with the IRC and expected German scholars to first join the International Mathematical Union (IMU), which belonged to the IRC, before attending the Congresses, from which the Germans had been excluded in Strasbourg (1920) and in Toronto (1924). Some of the German scientific bodies and scholars, in particular the Berlin Academy of Sciences, the German Mathematicians’ Association (DMV), and the Berlin mathematicians Ludwig Bieberbach, Erhard Schmidt, and Richard von Mises, opposed seeking admission to the IRC. These three mathematicians also refused to go to Bologna (a decision that partly backfired against their own work),15 both out of what they considered national pride and also resentment toward the past policies of the IRC.16 The boycott had indeed been considered merely destructive and politically one-sided by many mathematicians, not only those from Germany but also those from neutral Scandinavian countries, among others (Dauben 1980). The Italian organizers of the Bologna Congress, in particular the thenpresident of the IMU, Salvatore Pincherle (1853-1936), succeeded in loosening the contacts between the IRC (and thus the IMU) and the Congress. Hilbert and like-minded

6 Rightly, these three words (‘‘und ho¨chsten Ho¨he’’), which are redundant and stylistically ugly, are not reproduced in Reid’s partial translation mentioned at the end of this article (Reid 1970: 188). 7 In Reid (1970: 188) instead: ‘‘of the cultivation of the exact sciences.’’ 8 These three words are added in Reid (1970: 188). 9 The three words ‘‘oder gar Gegensa¨tze’’ are not included in Reid (1970: 188). They are, however, important, as explained below. 10 The German plural noun ‘‘Gru¨nde’’ can mean either ‘‘explanations/justifications’’ or ‘‘reasons.’’ Hilbert’s formulation is ambiguous here, in particular because ‘‘fadenscheinig’’ (literally ‘‘threadbare’’) can mean ‘‘easily seen through’’ (which fits to both interpretations of Gru¨nde) or ‘‘shabby,’’ with malicious intent, which fits better with ‘‘reasons.’’ 11 ‘‘Elemental’’ is my translation of the German ‘‘urwu¨chsig,’’ which has the connotation of ‘‘natural and uncivilized’’ and was therefore probably slightly undiplomatic in relation to Russian participants of the Congress. 12 See C. Goldstein’s discussion in Goldstein, et al. (2007: 383 ff.). 13 Apparently, by referring to rather thin ‘‘threads of twine,’’ Hilbert wanted to express that he did not consider the obstacles insurmountable. 14 The formulation ‘‘zu Stande gebracht’’ has the connotation of ‘‘against odds,’’ acknowledging the political efforts of the Italians. 15 Bru (2003: 176) allows the conclusion that the ICM in Bologna 1928 was a missed opportunity to establish connections between parallel French (Hadamard, Fre´chet), Czech (Hostinsky´), and German/Austrian (von Mises) work on Markov chains. Von Mises remained unaware of the presentations given there for several years. Conversely, the French did not learn about progress that von Mises had made with the help of the theory of positive matrices (G. Frobenius). 16 Apparently some institutional conflict between Berlin and Go¨ttingen played a role here as well. There was a rejection of modernist Go¨ttingen attitudes in communication (Zentralblatt!) by some Berlin mathematicians, coupled with jealousy about the Rockefeller funds that Go¨ttingen received in the 1920s. Cf. Siegmund-Schultze (2001).

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German mathematicians won the battle against nationalistic resentment in their own country and persuaded the majority of their countrymen of the good intentions of the Italians.17 Of course one may question the possible nationalistic motives of some Italians for holding the congress in Bologna, and whether Fascist Italy was the right place for Hilbert to talk about ‘‘mathematics and race.’’ By adding the Roman numeral VI to 1928 on the title page, the anonymous Italian editors of the Proceedings acceded to the official requirement to honor the respective (here: sixth) year of Fascist rule in Italy (Fig. 2).18 In fact the Fascist government heavily subsidized the congress. It appears, however, that Italian mathematicians had some reason at the time to believe in a prosperous development of Italian mathematics under Mussolini, whom, according to the Proceedings (Atti 1929) and to Tonelli (1929), they had asked to be the honorary president of the congress.19 The Jewish organizer of the ICM, Pincherle, who had joined the Fascist party after receiving support,20 certainly did not anticipate the anti-Jewish Italian legislation of 1938. All these historical facts have been described by historians previously21 and have been documented in, for instance, letters between Hilbert and his former student Bieberbach in advance of the congress. Some of these documents are even reproduced in the Proceedings (Atti 1929) of the Bologna Congress. The Proceedings quote in particular (in the original German) one leaflet by the Dutch topologist L. E. J. Brouwer (1881-1966) from January 1928, which warns the Germans against taking part in Bologna: ‘‘Each mathematician should ponder for himself whether participation in the planned congress is possible without mocking the memory of Gauss and Riemann, the cultural meaning of the science of mathematics and the independence of the human spirit.’’22 Publishing Pincherle’s letter to Picard, dated 8 June 1928 (Atti 1929: 5-10), in which he announces the severance of the congress from the IRC, the Proceedings emphasize the return of the Germans to international collaboration as the major political event connected to Bologna. Some historians and mathematicians have exaggerated the role of the Germans, in particular of David Hilbert, in Bologna. In 1964, Richard Courant, aged 76, an admirer of

Figure 2. Title page of volume 1 of the Proceedings of the Bologna Congress.

Hilbert and his former student who was present at Bologna, claimed to remember the following: ‘‘The allied countries wanted to make a gesture of conciliation and invited the German mathematicians and asked Hilbert to be the president of the congress in Bologna. Hilbert accepted but a group of nationalists became excited, and it all became a question of honor.’’ (Courant 1981:162) It is important to stress that even the testimony of participants such as Courant is not necessarily indisputable when made so many years after the event.23 Contrary to Courant’s memory, the organizer Pincherle was rightfully elected as the president of the congress and Hilbert became one of ten vicepresidents, one each for the ten major participating countries. If there was one foreigner who, indeed, received more

17 There was, however, some tactlessness on the part of the Italians who organized an excursion of the congress to newly acquired Italian territory at the Lago di Ledro, which until 1918 had belonged to Austria. Bieberbach pointed this out in a letter dated 18 June 1928 to the rector of Halle University, which was then made public against Bieberbach’s will. Bieberbach also claimed that the Lago di Ledro belonged to German-speaking South Tyrol (Van Dalen 2013:544). However, I should add that Bieberbach had got it partly wrong, because the lake is located in the Italian-speaking province of Trentino, also newly occupied by the Italians. 18 It should not be interpreted (as has been done) as retrospectively excluding the boycott congresses at Strasbourg (1920) and Toronto (1924) from the list of truly ‘‘international’’ congresses, which otherwise could be considered as the sixth and seventh ICMs. In fact, the ‘‘fifth ICM’’ in Cambridge 1912 was the last to be officially numbered, the Great War and the resulting problems in internationalism would end this tradition. 19 ‘‘The congress was held under the distinguished patronage of H. M. the King of Italy, and the honorary presidency of His Excellency, Benito Mussolini, the leader of the government’’ (Tonelli 1929: 201). 20 On the subsidies and Pincherle’s relation to the Fascists I gratefully received information from Annalisa Capristo (Rome), who is planning to publish a manuscript about the political aspects of the organization of the Bologna Congress, in particular the relationship between French and Italian mathematicians. 21 Dauben (1980), Lehto (1998), Mehrtens (1987), Siegmund-Schultze (2011), Van Dalen (2013). 22 Atti (1929: 10), my translation from German. 23 I return to this point at the end of this article.

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of a podium in Bologna than the others, it was the leading American George David Birkhoff.24 According to the Proceedings, Birkhoff not only thanked his Italian hosts with a few words at the opening session ‘‘in the name of the foreign Congress members’’ (Atti 1929: 74), he was also given the honor of presenting his plenary talk ‘‘Quelques e´le´ments mathe´matiques de l’art’’ on 10 September in the splendid environment of the Palazzo Vecchio in Florence (to which the congress had moved) as the last presentation of the entire Congress (Atti 1929: 85/86). The role of Birkhoff in Bologna resonated well with the increasing international importance of American mathematics in the 1920s and with the financial muscles of the Rockefeller philanthropy (advised by Birkhoff), which between 1926 and 1929 financed the new mathematics institute buildings in Paris and Go¨ttingen and (after 1924) had sponsored international fellowships (Siegmund-Schultze 2001 and 2015). What do we really know about Hilbert’s appearance in Bologna? Otto Blumenthal, Hilbert’s student and the longstanding managing editor of Mathematische Annalen, described the appearance of ‘‘frail’’ Hilbert and the ‘‘unanimous applause’’ for him (Blumenthal 1935: 427). But he connected the scene to Hilbert’s mathematical plenary talk in Bologna, ‘‘Problems of laying foundations for mathematics’’ (Probleme der Grundlegung der Mathematik) (Hilbert 1928), which concerned Hilbert’s logical proof theory. This agrees with the account of Hasso Ha¨rlen (1903-1989), a German mathematician born in Flemish Antwerp, who had been something like Brouwer’s unofficial envoy in Bologna. Ha¨rlen described Hilbert’s mathematical plenary talk and its reception by the audience in a letter to Brouwer after the Congress: ‘‘The first talk by Hilbert, who is greeted with a storm of applause. Frequent repetitions; his ability to concentrate clearly much influenced by physical suffering. Contents essentially known from recent publications. Great applause. Hadamard is also greeted with great applause, and his talk is also very good in presentation—much more effective than the one of Hilbert. With Hadamard the applause afterwards was much stronger than beforehand. With Hilbert the applause was almost only for the person, with Hadamard also for the talk.’’25

Reflections on Hilbert’s Intended Talk Looking at the draft of Hilbert’s address in the Go¨ttingen library in its entirety, there is no doubt that the French–German conflict of the past and reconciliation between French and German mathematicians was foremost on Hilbert’s mind in 1928. This follows, for instance, from the names of the important mathematicians in the passage immediately after the remark about mathematics and race (see my translation

above). Of these 30 mathematicians, 14 are French and 9 German. Thus the names were certainly not ‘‘thrown wildly (and randomly) between the nations.’’ Except for Abel, Hamilton, Huygens, and Newton, and, a bit further below, P. L. Chebyshev (Ka¨the Hilbert writes ‘‘Tschebischew’’), all were French- or German-speaking. Hilbert was, of course, aware of the nationally organized educational systems as a decisive background for the cultivation of mathematics in modern societies and was also aware that therefore a fully ‘‘random’’ list (in the sense of throwing dice) could not be expected. But even granted this social condition, Hilbert’s list was still biased, neglecting mathematicians from other mathematically strong nations. First, for Hilbert to present the draft (although Hilbert admits ‘‘superficiality’’ in the passage with the names) in Bologna in unchanged form would have been most embarrassing because it omits any mathematician from the host country Italy. To be sure, Italian mathematics reached its height later in history than French and German mathematics, and Hilbert did try to emphasize the Italian contribution later in the talk. However, Hilbert could easily have mentioned Felice Casorati (1835-1890) or Ulisse Dini (1845-1918) or other important deceased26 Italian mathematicians who were certainly no less important than, let us say, the French Gabriel Lame´ (1795-1870) or the German August Ferdinand Moebius (1790-1868). Further, one would have expected him to mention a man such as the Indian Srinivasa Ramanujan (18871920) as proof of the equal distribution of mathematical talent among the nations and peoples. In addition, and although Hilbert addressed only ‘‘gentlemen’’ in his talk, he could have pointed to Sophie Germain (1776-1831) or Sofia Kovalevskaja (1850-1891) as examples that mathematical talent also exists in women.27

Why Allusion to ‘‘Race’’? If the draft of Hilbert’s talk was about the reconciliation between ‘‘peoples,’’ in particular the French and the Germans, why does he refer to ‘‘races’’ as well? Admitting from the outset that speaking about ‘‘race’’ before the Nazi crimes occurred was certainly less burdened with connotation than today and often just referred to national peculiarities, we need nevertheless to obtain more historical background to answer this question.28 There was of course, even in the late 1920s, academic anti-Semitism in Germany, although less so than under the monarchy before 1914 (Siegmund-Schultze 2008). Discrimination against Jewish mathematicians in academic appointments, even those who had converted to the Christian faith, often had xenophobic overtones, particularly if the candidates were immigrants to Germany (as in the case of Chaim Mu¨ntz, described later). But even

24

However, Birkhoff was not elected vice-president of the Congress for the United States, but rather Oswald Veblen. This underlines the growing strength of American mathematics at the time. 25 H. Ha¨rlen in a letter to Brouwer, written in German and dated 27 September 1928 (Van Dalen 2011: 335/336). Ha¨rlen’s appearance in Bologna and his anti-Bologna and anti-French propaganda there were noted with disapproval by the French mathematician Jacques Hadamard, who criticized Brouwer’s indirect actions in Bologna (through Ha¨rlen) in a letter to Albert Einstein, dated 16 October 1930 (Einstein Archives Jerusalem, 13-176). 26 It seems that in his talk he did not want to refer to living mathematicians. 27 In Go¨ttingen Hilbert continuously supported Emmy Noether (who was also at the ICM in Bologna) against gender discrimination, showing that he recognized the importance of the issue for the flourishing of mathematics. 28 For those who read German, much of this background can be gathered from Mehrtens (1990). Others may resort to Forman (1971).

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Jewish mathematicians born in Germany often faced resistance from nationalist professors. We read in a letter concerning candidates for a teaching position, from Richard Courant to the topologist Helmuth Kneser in Greifswald, dated 9 January 1928, that is, the year of the Bologna Congress: ‘‘Allow me one remark about the one condition—Aryan descent—connected by your colleagues to the teaching position. I cannot understand why—when objective interests demand the opposite—one should give in to the maliciousness, bigotry, and stupidity of individual colleagues. … This is diametrically opposed to the interests of science and of the university… I gather that the presence of any of the three colleagues proposed [Hans Lewy, Stefan Cohn-Vossen, Chaim Mu¨ntz] would greatly contribute to the ennoblement [Veredelung] of the views of parts of your faculty.’’29 Despite Courant’s protests, the remunerated teaching position in question went instead to Wilhelm Su¨ss (18951958), who later, under the Nazis, was for many years president of the German Mathematicians’ Association, when these three Jewish mathematicians had had to leave the country (Siegmund-Schultze 2009). Even an unpaid teaching position (based on Habilitation) for Kurt Mahler (1903-1988), the noted number-theorist born in German Krefeld, was ruled out in the same University of Greifswald five years later. This is documented in another letter, this time written by Kneser to Courant on 1 December 1932, two months before the Nazis came to power: ‘‘A group of our faculty is more or less anti-Semitic and would oppose Mahler’s Habilitation with all means. … So I must advise Mahler against this plan. I do this with the greatest regret because I have the highest respect for his ability. Even Reinhardt had to admit this much, however he spoke strongly against the Habilitation, without giving reasons.’’30 Hilbert knew of incidents such as these even in Go¨ttingen around 1900, although he and Felix Klein (1849-1925) had, for the most part, managed to circumvent existing antiSemitic resentment in the Philosophical faculty in appointment policies (Rowe 1986). Hilbert knew, of course, that xenophobic and anti-Semitic propaganda often came with auxiliary ‘‘arguments’’ concerning the ‘‘style’’ of mathematics preferred by the rejected individual, because one could not possibly deny the mathematical prowess of certain candidates.31 For instance, controversial discussions about the ‘‘Landau style’’ in mathematics occurred long before 1933 (Rowe 1986: 438; Siegmund-Schultze 2012: 34). Not coincidentally we find the following three words in a

passage of Hilbert’s draft, which I emphasize: ‘‘It is a complete misunderstanding of our science to construct differences or even incompatibilities [oder gar Gegensa¨tze] according to peoples and races, and the reasons for which this has been done are very shabby ones.’’ Hilbert must have been aware that anti-Semitism before and during the War was part of a much broader obsession with race and national differences in culture and even in intellectual abilities, including talent for sciences. In the 19th century some German (Eugen Du¨hring) and French (Arthur de Gobineau) philosophers had developed a racist, as opposed to religious, notion of Jewishness. Somewhat later, the international eugenics movement (in both its rightist and leftist versions) stimulated, used, and misused pedagogical, psychological, and anthropological investigations, interpreting them with racist overtones. Even Felix Klein, for example in his Go¨ttingen seminar ‘‘Mathematics and Psychology’’ of 1909, was actively involved in discussions of race and mathematics. The Jewish statistician of Go¨ttingen, Felix Bernstein, who took part in this seminar, dedicated some of his statistical research to the investigation of racial peculiarities.32 In broader circles similar investigations were supported and used to ‘‘explain’’ otherwise unexplainable developments of the modern social world. I have no information about Hilbert’s attitude toward Klein’s pedagogical and psychological seminars. It seems likely that Hilbert, in hindsight after the War, saw a connection between those not ‘‘strictly mathematical’’ investigations and the nationalist propaganda during the War, culminating for instance in the manifesto of the 93 ‘‘To the Civilized World’’ (‘‘An die Kulturwelt’’). This infamous appeal, dated 4 October 1914, was signed by 93 ‘‘representatives of German scholarship and art,’’ among them Felix Klein, the artist Max Liebermann (who created a famous portrait of Klein), and the physicist Max Planck, but David Hilbert did not sign.33 Here for instance we find the following use of ‘‘race’’ in the context of War propaganda: ‘‘Those who have allied themselves with Russians and Serbians, and present such a shameful scene to the world as that of inciting Mongolians and negroes against the white race, have no right whatever to call themselves upholders of civilization. … Were it not for German militarism German civilization would long since have been extirpated. … The German Army and the German people are one.’’ (To the Civilized World 1919: 285) Before and during the War it was not just the German side that tried to make a political point of psychological and pedagogical reflections about race and nation. Hilbert was certainly aware of the French campaign during WWI against German mathematicians and the style of German

29 State and University Library Go¨ttingen, Manuscript Division, cod Kneser-H-A-15-1-Courant. Fol.13. Translated from German. The ‘‘Veredelung’’ is apparently an ironic protest against ‘‘primitive’’ views (in Courant’s opinion) in the faculty. 30 Ibid. Fol. 21. Translated from German. Karl Reinhardt (1895-1941) was at the time professor of mathematics in Greifswald. 31 Even the anti-Semite Reinhardt in the example quoted from Greifswald (1932) reluctantly admitted the mathematical strength of Mahler. 32 Chislenko and Tschinkel (2007) mention Felix Bernstein and Hermann Weyl as participants in Klein’s seminar. 33 Reid (1970: 137/138) claimed without documentation that Hilbert refused to sign the manifesto. However, I assume Hilbert was not even asked because the initiators of the manifesto knew about his lack of nationalist fervor. Anyway, to have one signature from a Go¨ttingen mathematician (Klein) was probably enough for the initiators.

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mathematics, epitomized by writings such as those by the important specialist in classical mechanics and philosopher of physics Pierre Duhem (1861-1916) and by the famous function-theorist E´mile Picard, who later led the IRC. In ‘‘Quelques Re´flexions sur la Science Allemande’’ (1915), Duhem said, partly borrowing terminology from Pascal (Kleinert 1978: 516/17): ‘‘The German spirit [l’esprit allemand] is basically geometrical. The German does not have the esprit de finesse. … The geometrical spirit gives shape to a building [donne corps a` l’e´difice]34 which has been constructed [conc¸u] before by the [French] inventive spirit [esprit de finesse].’’35 And in ‘‘L’histoire des sciences et les pre´tentions de la science allemande,’’ Picard added in 1916: ‘‘It is a strange aberration that the German race claims to be the only one in the world to contribute to the scientific development of mankind. It is a collective dementia which pushes the German people like a chosen people charged by its God to direct the world.’’36 Although the Catholic Picard probably had primarily the Protestant god of the Prussians on his mind, using the formulation with ‘‘a Chosen people’’ has, in my opinion, clear anti-Semitic overtones. Picard was certainly aware of the fact that several leading German mathematicians had Jewish ancestry (apparently more so than the French, with Jacques Hadamard being the most important mathematician with Jewish background). In any event there is much anti-Modernist sentiment in Picard’s quote and in many of his other writings, in particular a fear of globalization and modernization of mathematics. After the War and throughout the 1920s nationalist resentment persisted among various European scholars and students, which in Germany was directed toward an undoing of the results of World War I. Also in Go¨ttingen anti-Semitic actions and, beginning in the late 1920s, National Socialist activities, particularly in the student body, increased political tensions. At the same time, German nationalist mathematicians such as the future Nazi functionary Theodor Vahlen began talking about differences in mathematics that were conditioned by the ‘‘races’’ of mathematicians. In a published talk of 1923 Vahlen called mathematics a ‘‘mirror of races’’ (Vahlen 1923: 22). Notoriously, various streams of ‘‘Lebensphilosophie,’’ most influentially represented by Oswald Spengler’s ‘‘Decline of the West’’ (1919, first English translation in 1928), criticized the allegedly one-sided insistence of the sciences and mathematics on causality and logical thinking and contained various remarks about the influence of human ‘‘races’’ on cultural and scientific production (Forman 1971).

Insecurities and Foundations Hilbert and other modernist and internationalist mathematicians saw parallels between political insecurity and problems in the foundations of mathematics. Political feelings may have exaggerated concerns about a possible anti-Cantorian backlash in the logical foundations of mathematics, not least because the principal opponent of mathematical formalism, the Dutch intuitionist and topologist L. E. J. Brouwer, who was Bieberbach’s ally in the anti-Bologna movement, seemed to personify both the mathematical and the political counterrevolution (Mehrtens 1990; Van Dalen 2013). It is clear from many of his actions (support for liberal and leftist mathematicians and philosophers such as Emmy Noether and Leonard Nelson) that Hilbert was no ‘‘conservative’’ within the broad political spectrum of the Republic of Weimar. However, one cannot claim either that Hilbert in his social and cultural views was a clear-cut ‘‘progressive.’’ Indeed I will quote below his student Max Born, who seems to claim the opposite, although, in my opinion, he exaggerates. Hilbert can probably be best described as a strong individualist who held rather unconventional, but mostly liberal views.37 But in any case he had a very good feeling for those ideologies and political opinions outside mathematics that began to threaten mathematical communication itself. In publications before Bologna, Hilbert had used strong political vocabulary in a 1922 article against Brouwer’s intuitionism. ‘‘Brouwer is not … the revolution, but only a repetition, with the old tools, of an attempted coup [Putsch] that, in its day, was undertaken with more dash, but nevertheless failed completely; and now that the power of the state has been armed and strengthened by Frege, Dedekind, and Cantor, this coup is doomed to fail.’’ (Hilbert 1922: 1119) In his Ko¨nigsberg talk of 1930, published in ‘‘Die Naturwissenschaften,’’ Hilbert alluded to Spengler, after he had quoted Jacobi’s remark that ‘‘the sole aim of all science is the honor of the human spirit’’ in his famous letter to Legendre of 1830: ‘‘Whoever feels the truth of the magnificent manner of thinking and of the world-view that shines forth in these words of Jacobi will not fall into retrogressive and fruitless skepticism; he will not believe those who today, with a philosophical air and a superior tone, prophesy the downfall of culture and fall into an ignorabimus. For the mathematician there is no ignorabimus, nor, in my opinion, for any part of natural science.’’ (Hilbert 1930: 1165) In Bologna in 1928, striving for reconciliation with the French, Hilbert toned down his fears, at least as far as the foundations of mathematics were concerned. In his plenary talk (Hilbert 1928), described previously in the report by

34 One may interpret this as a discussion of the relationship between ‘‘form and content’’ in mathematics, which was reinvigorated later with the advent of ‘‘modern, structural’’ mathematics. Remarkably, ‘‘geometry’’ is not in Duhem’s quote associated with ‘‘intuitive,’’ organic, traditional mathematics but with the modern logical tendencies, as for example represented at the time by Hilbert’s ‘‘Grundlagen der Geometrie’’ (1899). In the antimodernist Nazi propaganda by L. Bieberbach and other Germans 20 years later, ‘‘logic’’ is opposed to ‘‘intuitive’’ geometry, and French and ‘‘Jewish’’ mathematics are accused of relying too much on purely logical spirit (Mehrtens 1987). 35 Not accidentally, and in opposition to sweeping judgments such as the one by Duhem, Hilbert, in the draft of his political speech, stressed that the origin of modern geometry lay in France in the 17th century, to be followed later by collaboration between the French and the Germans. 36 My translation of Picard (1916: 21) from French. 37 Reid (1970) provides entertaining anecdotal evidence for Hilbert’s ‘‘originality,’’ which sometimes bordered on eccentricity.

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Ha¨rlen, Hilbert did not mention Brouwer at all. He made small sideswipes at Poincare´ and Bertrand Russell in this talk and reiterated his famous ‘‘In mathematics there is no ignorabimus,’’ already known from his talk at the ICM in Paris in 1900. In the draft of his political talk we have seen that Hilbert called Poincare´ the ‘‘most splendid representative of mathematics in an entire period’’ and referred to Brouwer only indirectly by stating confidently that ‘‘we won’t be irritated by isolated dissenters.’’ However, in Bologna Hilbert must have been aware, for instance from the plenary talk given there by the Russian Nikolaj Luzin, one of the vice-presidents of the Congress, that there were mathematicians around who shared some of Brouwer’s concerns with respect to an unlimited use of set theory.38 Of course, Hilbert’s proof theory of the 1920s was in itself partly a response to intuitionism, acknowledging some of its objections. Luzin in his talk ‘‘Sur les voies de la the´orie des ensembles’’ (On the ways [methods] of set theory) alluded to Hilbert’s program and called it ‘‘the biggest event in mathematics’’ (le plus grand e´ve´nement dans les mathe´matiques) (Lusin 1928: 298). However, Luzin made in the same talk a skeptical allusion to Hilbert’s famous recent (1925) statement,39 ‘‘No one shall be able to drive us from the paradise that Cantor created for us’’: ‘‘It is in addition the fatigue of ‘Cantor’s paradise’ which plays a certain role: an enormous amount of new notions without applications and still40 without contact with other branches of classical mathematics, many works on clearly artificial subjects and at least out of proportion with respect to their usefulness—all this has required extreme caution.’’41 The leader of the Russian ‘‘semi-intuitionistic’’ school of the theory of functions went on, praising the Dutch scholar who had just argued against the participation of the Germans in Bologna: ‘‘The success of the theory of Mr. W. [sic] Brouwer has its real basis not in the courageousness of the famous scholar (although one should be infinitely courageous in these matters) but in the artificial investigations just cited which are based on the notion of actual infinity.’’42 In view of presentations such as this one by Luzin, which was far from declaring Hilbert the winner in the struggle over the foundations of mathematics, it seems likely that 38

Hilbert’s two manuscripts in Bologna, the mathematical and the political, do not fully show the double concerns he had at the time, which were probably further aggravated by his health situation. In his ‘‘Lebensgeschichte’’ (Hilbert’s life history), Blumenthal writes that while Hilbert was in Bologna he had a serious setback in his life-threatening pernicious anemia from which he had suffered for several years.43 It seems likely that this setback in health increased his feeling of insecurity also in mathematics.

The Aftermath of the Bologna Congress: Hilbert’s Intended Warning Remains Unheeded According to Van Dalen (2013: 564), while Hilbert was at the Bologna Congress he talked to Blumenthal about his plans to remove Brouwer (and Bieberbach) from the editorial board of the Mathematische Annalen. His main reason was his fear about the future of modern set-theoretic mathematics. The removal was effected immediately after Bologna with the help of Blumenthal and others. Einstein, who had been on the board of the Annalen too, spoke about the affair in ironic terms as ‘‘The Battle of Frogs and Mice’’, alluding to a pseudo-Homeric fable (Van Dalen 2013: 552-588). Until the time of the Bologna Congress, Bieberbach—the later proponent of ‘‘Deutsche Mathematik’’—had still been widely considered a political liberal and supporter of the Weimar Republic. This impression began to appear as an illusion now, as the Go¨ttingen physicist Max Born revealed. The famous cofounder of modern quantum mechanics was Hilbert’s former student and had taken part in Bologna. In a letter to his friend Einstein, dated 20 November 1928, that is, after the Bologna congress, Born said: ‘‘Hilbert is not politically very left-wing; on the contrary, for my taste and even more for yours, he is rather reactionary.44 But when it comes to the question of the intercourse between scientists of different countries, he has a very sharp eye for detecting what is best for the whole. Hilbert considered, as we all did, that Brouwer’s behaviour in this affair, where he was even more nationalistic than the Germans themselves, was utterly foolish. But the worst of it all was that the Berlin

However, observers of the congress such as the Frenchman Adolphe Buhl seemed to consider Luzin an ‘‘outsider,’’ too. Buhl managed not to mention Luzin at all in his report on the Congress, although he discussed all other plenary lectures. Although Buhl admitted that not all mathematicians might follow Hilbert’s optimism about the solvability of all problems (Buhl 1928: 195), Buhl then said: ‘‘The proposals uttered in Hilbert’s lecture have been confirmed by all (plenary talks) which followed’’ (Buhl 1928: 196). The French seem to have been divided in their opinion about Luzin’s approach to function theory. According to Luzin’s report of November 1929 (Ermolaeva 1989: 221-223), his famous book of 1930 (Lusin 1930) was proposed by Henri Lebesgue, when they met in Bologna. He completed the book during a Rockefeller-sponsored fellowship in Paris with Lebesgue, which started after the Congress and lasted until May 1930 (Ermolaeva 1989, Siegmund-Schultze 2001). 39 This was from Hilbert’s talk ‘‘On the Infinite,’’ translated in Van Heijenoort (1967: 376). 40 ‘‘Still’’ seems to be a more adequate translation of ‘‘toujours’’ here than ‘‘always.’’ 41 ‘‘D’ailleurs, c’est la fatigue du «paradis de CANTOR» qui a joue´ un certain roˆle: une quantite´ e´norme de notions nouvelles sans applications et toujours sans contact avec les autres branches des mathe´matiques classiques, les nombreux travaux aux sujets manifestement artificiels et du moins hors de proportion avec leur utilite´— tout cela exigeait une prudence extreˆme’’ (Lusin 1928: 298/99). 42 Ibid., p. 299: ‘‘Le succe`s de la the´orie de M. W. BROUWER a sa vraie cause non pas dans la hardiesse du ce´le`bre savant (bien que dans ces choses il faut eˆtre infiniment hardi), mais dans l’artifice de´ja` cite´ des conside´rations base´es sur l’infini actuel.’’ 43 Hilbert suffered from the same disease as the Norwegian Sophus Lie, who died from it in 1899. But Hilbert was finally cured by new American medicine, partly mediated by the Rockefeller Foundation (Blumenthal 1935: 427). 44 This characterization by Born has to be taken with a grain of salt, given Hilbert’s support for women in mathematics such as Emmy Noether and given also the text of his planned Bologna talk. Above all, Born’s words are meant as a comparison with Einstein, who was indeed much more publicly engaged and explicit in his political opinions than Hilbert.

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mathematicians were completely taken in by Brouwer’s nonsense. I would like to add that the Bologna business was not the decisive factor—only the occasion for Hilbert’s decision to remove Brouwer [from the Annalen; R. S.-S.]. I can understand this in Erhard Schmidt’s case, for he always did lean to the right in politics, as a result of his basic emotions. For Mises and Bieberbach, however, it is a rather deplorable symptom.’’ (Born/Einstein 1971: 98) A symptom of the times indeed it was. The mathematicians who gathered at Bologna agreed that there were still international political conflicts also in mathematics and they decided to reconvene in the politically neutral surroundings of Zu¨rich in 1932. The year 1929 saw the beginning of the Great Depression, and even the resourceful Rockefeller Philanthropy had to reduce its activities in Europe, focusing from then on even more on the United States (Siegmund-Schultze 2001). The seizure of power by the German Nazis in 1933 brought international mathematical communication for mathematicians within Germany almost to a standstill. Bieberbach, who had become a Nazi, would soon speak deprecatingly about ‘‘international formalism’’ in mathematics. At the same time, under the political apartheid conditions of Nazi Germany, alleged ‘‘differences’’ and ‘‘incompatibilities’’ between German and Jewish mathematics became a pretext for discrimination and dismissals (Segal 2003). In view of mass emigrations Hilbert may well have then uttered his famous remark to the Nazi minister of education: ‘‘There is no mathematics in Go¨ttingen anymore.’’ 45 Five years later (1938) anti-Semitic legislation was introduced in Italy as well; the organizer of the congress, Pincherle, who was of Jewish origin and died in 1936, was spared the persecution that then followed. Thus in the decade after Bologna the farsightedness (a ‘‘sharp eye’’ as Born said) of Hilbert’s unpublished appeal that ‘‘Mathematics knows [or rather should know] no races’’ and of Hilbert’s relentless internationalist outlook became obvious in Germany, in Italy, and worldwide. Bologna should remind us that reconciliation and normalization in the sciences and mathematics are not irreversible.

Postscript on Mathematics and Historiography This article is primarily about an episode from the life of the wonderful and still influential German mathematician David Hilbert. The article contains a hitherto unpublished and—at least in its full length—largely unknown political text authored by this mathematician. Hilbert certainly deserves a scientific biography, yet to be written, that distinguishes between facts and interpretation. My arguments have been partly interpretative, but I hope to have

indicated what can be documented and what is my interpretation. This episode from Hilbert’s life also provides an opportunity to discuss some issues of historiographic methodology. In fact the research performed for this article was triggered by Constance Reid’s book on Hilbert, in which she claims for a fact that Hilbert presented this political speech in Bologna (Reid 1970: 188), and she quotes a longer passage from his manuscript in English translation. She embellishes her description with some remarks about Hilbert’s appearance at the talk, which she apparently took from Blumenthal’s biography, however, which had alluded to Hilbert’s mathematical talk. Reid’s claim, like many other factual claims in her successful biographies of various mathematicians (Hilbert, Courant, Neyman), is not backed up with references to sources. Her description of Hilbert’s alleged presentation in Bologna has been quoted in almost all the historical accounts of the Bologna congress that appeared after her biography in 1970. It has been reproduced in several official histories of the ICM and the IMU and has been accepted as a fact even by serious historians.46 This may be because it was known that Reid had firsthand access to Hilbert’s papers, which lay then, in the 1960s, unexplored and unregistered in the Go¨ttingen Mathematical Institute.47 One may agree with Alexanderson (2011) that Reid’s research has great merits in alerting the public to Hilbert’s papers and that her research may have even saved Hilbert’s papers from possible destruction. Reid’s hybrid work on Hilbert, of course, does not pretend to be a scientific biography; it does not even include as much as a bibliography. But the book has been influential in shaping our image of Hilbert. It made a strong impression on me when I first began to study the history of mathematics. Recently the editor of a journal with wide circulation among mathematicians categorically refused to publish any criticism of Reid’s books. I received the impression he shared my opinion but was not able to convince angry refereemathematicians who felt that I lacked ‘‘respect’’ for Reid. In the end I had to withdraw my manuscript. Whence this influence of Reid’s books? First of all, and again: They are well written. They exhibit a good understanding of the spirit of the times and of the persons described. What is more, the books often contain reasonable mathematical judgment, maybe based on sound advice by Reid’s mathematician-sister Julia Robinson. One can certainly say that she improved the standards in popular mathematical biography compared to E. T. Bell, to whom, incidentally, she devoted a biography that is partly critical of Bell’s disregard for accuracy.48 I personally have no doubt that for many years to come Reid’s books will remain immensely valuable sources of inspiration and edification for mathematicians and historians alike. Sometimes they may serve as a stimulus to check and compare claims that have been made there. If there is

45

This is claimed without documentation in Reid (1970: 205). Albers et al. (1987: 20), Curbera (2009: 83), Lehto (1998: 47/48), even Mehrtens (1987: 215), Riehm/Hoffman (2011: 170), and Van Dalen (2013: 550). 47 Today, as mentioned previously, the Hilbert papers are in the Manuscript division (Handschriftenabteilung) of the Go¨ttingen University Library and are accessible for historical research. 48 I owe this remark to Marjorie Senechal. 46

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one reproach that could be made against Reid, then, it is that she did not warn her readers in the preface that many of her arguments, which relied on interviews with mathematicians such as Richard Courant and Kurt Friedrichs conducted nearly half a century after the fact, included necessarily a margin of error. It finally seems worth reflecting on the reason that mathematicians have mostly accepted Reid’s biographies at face value. Maybe they just do not expect a level of rigor in a historical report that would compare with mathematical standards. Moreover, although quite a few events are misrepresented in Reid’s books, most of the events could easily have happened. Just as in mathematics itself, some mathematicians might be more interested in the plausibility and consistency of alleged facts than in their relation to reality. Do there exist, in the end, differences between mathematical and historical truth? I hope this article can contribute to a discussion of this and similar problems of historiographic methodology.

Blumenthal, O. 1935: Lebensgeschichte, in: Hilbert (1935), 388-429. Born, M., and Einstein, A. 1971: The Born-Einstein-Letters, London, MacMillan. Bortolotti, E. 1928: Congresso internazionale dei Matematici (Bologna 3.-10.9.1928). Bollettino U. M. I. 7, 221-228, 266-284. Bru, B. 2003: Souvenirs de Bologne, Journal de la Socie´te´ Franc¸aise de Statistique, 144, 135-226. Buhl, A. 1928: Souvenirs de Bologne, L’Enseignement Mathe´matique 27, 193-202. Chislenko, E., and Tschinkel, Y. 2007: The Felix Klein Protocols, Notices AMS 54, 960-970. Courant, R. 1981: Reminiscences from Hilbert’s Go¨ttingen, The Mathematical Intelligencer 3, no. 4, 154-164. Curbera, G. P. 2009: Mathematicians of the World, Unite! The International Congress of Mathematicians—A Human Endeavor, Wellesley, A. K. Peters. Dauben, J. 1980: Mathematicians and World War I: The International Diplomacy of G. H. Hardy and Go¨sta Mittag-Leffler as Reflected in Their Personal Correspondence, Historia Mathematica 7, 261-288. Ermolaeva, N. S. (ed.). 1989: Correspondence between N. N. Luzin

ACKNOWLEDGMENTS

This article is an extended version of parts of an invited talk presented at the History of Mathematics Section of the ICM in Seoul, South Korea, on 19 August 2014. For other parts of the talk, see my article (2014) in the Proceedings of the Congress. I thank June Barrow-Green for critical discussion of the manuscript and help with the English. Marjorie Senechal suggested disentangling the story of Hilbert from my criticism of Reid. I thank the Handschriftenabteilung Niedersa¨chsische Staats- und Universita¨tsbibliothek Go¨ttingen (Hilbert Papers, Kneser Papers) and the Einstein Archives of Jerusalem University (Einstein Papers) for permission to quote material in their possession. Annalisa Capristo (Rome), Karine Chemla (Paris), Ulf Hashagen (Munich), Tilman Sauer (Bern), Norbert Schappacher (Strasbourg), and Dirk van Dalen (Utrecht) provided useful responses as well. Faculty of Engineering and Science University of Agder Gimlemoen, Postboks 422 4604 Kristiansand S Norway e-mail: [email protected]

and A. N. Krylov (Russian), Istoriko-Matematicheskie Issledovanija 31, 203-272. Ewald, W. (ed.). 1996: From Kant to Hilbert. A Source Book in the Foundations of Mathematics, 2 volumes, Oxford, Clarendon Press. Fehr, H. 1929: Le Congre`s de Bologne 3-10 Septembre 1928, L’Enseignement Mathe´matique 28, 28-53. Forman, P. 1971: Weimar Culture, Causality, and Quantum Theory, 1918-1927: Adaptation by German Physicists and Mathematicians to a Hostile Intellectual Environment, Historical Studies in the Physical Sciences 3, 1-115. Gingrich, C. H. 1928: International Congress of Mathematics at Bologna, Popular Astronomy 36, 529-532. Goldstein, C., Schappacher, N., and Schwermer, J. (eds.). 2007: The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae, Berlin, Springer. Hilbert, D. 1922: Neubegru¨ndung der Mathematik. Erste Mitteilung (1922), here quoted from the English translation in Ewald (1996, volume II, pp. 1115-1134). Hilbert, D. 1928: Probleme der Grundlegung der Mathematik, in Atti (1929, vol. 1, pp. 135-141). Hilbert, D. 1930: Naturerkennen und Logik, here quoted from the English translation in Ewald (1996, volume II, pp. 1157-1165). Hilbert, D. 1935: Gesammelte Abhandlungen, vol. 3, Berlin, Springer. Kleinert, A. 1978: Von der Science allemande zur Deutschen Physik:

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