At frst it looked like European politics were moving in the right direction all by themselves. The bundle of treaties negotiated in 1925 at Locarno, principally by Germany's Foreign Minister Gustav Stresemann(1878–1929), Aristide Briand (1862–1932) from France, and Austen Chamberlain (1863–1937) from Great Britain guaranteed the German borders with France and Belgium and opened the way for Germany to adhere to the League of Nations. Germany was admitted as a new member to the League in September 1926, thereby bringing the treaties into efect. The same year Stresemann shared the Nobel Peace Prize with Aristide Briand.

Even before this, on 29 June 1926, the Assembly of the IRC decided to invite Germany, Austria, Bulgaria, and Hungary to join the Council and its subordinate scientifc organizations. The politics of exclusion which had governed the very foundation of the IRC and the IMU thus appeared to have come to an end, and preparations of the 1928 ICM proceeded accordingly. The IMU followed its president and formally decided on Bologna as the location of the Congress,120 and the other organization presided over by Pincherle:

the Unione Matematica Italiana, on which fell the heavy task of preparing the Congress, decided in view of the new situation to take up the traditions of the International Congresses from before the war, and lift all exclusions depending on political considerations. The Congress was placed under the auspices of the University of Bologna. Its organization was put into the hands of a local committee consisting of professors from this university and distinguished citizens.121

Recall that Pincherle was professor at Bologna; he himself took the chair of the committee; his principal helping hand would be his Bologna colleague Ettore Bortolotti (1866–1947). The patronage of the Congress was extended to his majesty the King of Italy. The *Capo del Governo*, i.e., Mussolini, was declared Honorary President of the Congress.

However, as soon as the announcements of the Congress were sent, in their thousands, "to all parts of the world where a mathematical school existed", opposition arose, revealing that the attempt to return to prewar routines was premature. Actors on either side could no longer pretend to have overcome the political legacies of the war and its aftermath.

119 One can still get a lively impression of the tensions Pincherle had to negotiate in the concise account he published in the Bologna Proceedings; see Proceedings ICM 1928, vol. I, pp. 5–10. More recent studies include [Capristo 2016], [Giacardi & Tazzioli 2021].

120 Rather than Stockholm. The place of the 1928 Congress had been left open in 1924. Cf. [Lehto 1998], p. 44.

121 See Proceedings ICM 1928, Vol. I, pp. 5–6; my translation.

**Fig. 4.7** Salvatore Pincherle about 1900.

#### **<IP_ADDRESS> The German Reaction**

The frst objections came from a certain faction of the German mathematical community. In reaction to the disastrous outcome of World War I and the ensuing boycott of German science, a majority of German scientists developed a *de facto* political attitude which has attracted the attention of historians (of science), not least because of its peculiar mixture of partially contradictory principles. Viewing German science as one of the last remaining assets of the defeated nation was thus mixed with contempt for the new democratic regime; the fundamental conviction of the transnational and thereby international character of scientifc knowledge went hand in hand with a refusal to engage in certain international cooperations; pleas for the apolitical nature of science could go along with the explicit consideration of national interests in scientifc matters.122

In the early 1920s there was a German project of a blacklist for all foreign scientists backing the boycott of German science. The Göttingen number theorist Edmund Landau (1877–1938) was invited to contribute to this project and identify mathematicians who should be put on the list. He feared the difculties of coming up with sufciently reliable information about his foreign colleagues, considering that the whole project had to be "realized as seriously and carefully as it would necessarily be done by German scholars." Nevertheless, he did agree to provide specifc information when asked about individual cases.123

A serious political cleavage of the German mathematical community became apparent during the second half of the 1920s on two diferent, yet related scenes: within the editorial board of the journal *Mathematische Annalen* and in the fght about whether or not to follow the invitation to Bologna. In both cases the Dutch and violently anti-French fgure of Luitzen Egbertus Jan Brouwer led the camp of German nationalists. Siding with him faithfully in both fghts was the Berlin mathematician Ludwig Bieberbach (1886–1982), who would afterwards turn into the infamous leader of a Nazi variant of *Deutsche Mathematik*, based on a racist theory of mathematical creativity.

The rows within the editorial board of *Mathematische Annalen* started in 1925, when a contribution by Painlevé, proposed by Einstein, to a special volume dedicated to the memory of Riemann was refused in view of Painlevé's utterances during the war. The fghting was cut short in October 1928—shortly after the Bologna Congress—when David Hilbert single-handedly fred Brouwer from the editorial board "in view of the incompatibility of our views on fundamental questions." In this *Annalen afair*, political diferences merge with controversies about the foundations of mathematics, and the dramatic fnal coup was also related to Hilbert's health problems at the time.124

Insofar as the IRC had from its very foundation been the symbol of anti-German science politics, it is not surprising that it continued to function as a red rag for a majority of German scientists even when the Central Powers had been invited to join in 1926. As a matter of fact, much to the chagrin of German foreign policy makers in the Weimar Republic, the opposition of scientists across all felds would block Germany from joining the International Research Council, and the deadlock persisted even when, in 1931, the IRC transformed into ICSU, the *International Council of Scientifc Unions*, whose chief diference with the IRC was that it accepted scientifc unions as full members, on a par with nations.125

122 This paragraph is freely adapted from [Schappacher & Kneser 1990], p. 54.

123 See [Corry & Schappacher 2010], pp. 437–438.

124 The richest account to date of the *Annalen afair*is available in [Rowe & Felsch 2019]. In addition to the correspondence of Otto Blumenthal (1876–1944), the editor in chief of *Mathematische Annalen*, some of Brouwer's correspondence, in particular with Karl Kerkhof (1877–1945), is also reproduced and commented there.

125 Cf. [Schroeder-Gudehus 1966], pp. 255–265.

It was thus not so easy to smoothly return to a pre-war format of ICMs. The fact that the IMU—at the time still a subordinate structure of the IRC—was ofcially behind the invitations to the Bologna ICM in 1928 could raise suspicions about this invitation. National feelings in Germany and Austria could also be excited when Pincherle, despite countless exchanges with foreign colleagues, arranged to issue invitations only on behalf of the University of Bologna. One example of this was an excursion ofered to Congress participants on 7 September 1928 to Riva di Garda and its electrical power station, in the Province of Trento, which until World War I had been part of Austrian South Tyrol.

Brouwer and Bieberbach won over all their Berlin colleagues, notably Erhard Schmidt (1876–1959) and Richard von Mises, to refuse the invitation, and the *Deutsche Mathematiker-Vereinigung* also abstained from sending a delegation to Bologna. However, David Hilbert and the Göttingen mathematicians, as well as quite a few other German mathematicians did take part in the ICM. In this way the postwar politics of exclusion of German mathematicians would even serve to refuel the decade long competition between Göttingen and Berlin as mathematical centers in Germany. In the end, the mathematicians from Germany participating at the 1928 ICM were 75 men and one woman. The latter, a German mathematician, gave a lecture in the Number Theory section. In the title of her lecture she used her favorite word when thinking about mathematics: *Aufassung*—see Section 6.1.2 below.126

A story told by Constance Reid has it that when the German delegation led by David Hilbert entered the hall of the Congress, there was silence frst, followed by a standing ovation, and Hilbert delivered on the spot a short address which contained in particular the sentence: "Mathematics knows no races."127 A handwritten text of such a little speech has actually survived in Hilbert's papers related to Bologna. All this has been scrutinized in [Siegmund-Schultze 2016]; there is no evidence that Hilbert actually gave this speech. Nevertheless, since one has the handwritten notes, one may ponder this text, and in particular wonder about the fact that in 1928 Hilbert insisted not just on the fact that mathematics could transcend national borders, but on the question of race. The most likely explanation for this is the growing anti-semitism that Hilbert could observe in the 1920s, in Göttingen and elsewhere.128

#### **<IP_ADDRESS> The French Reaction**

German nationalists were not the only ones boycotting the would-be apolitical Congress at Bologna.

The rules established by the IMU in the immediate postwar period only allowed "scientifc groups belonging to member countries of the IRC" to be invited to the International Congresses. These rules—which were already breached at the Toronto Congress . . . —were now recalled, and in the most peremptory manner, when it became known that the invitations

126 The title was: *Hyperkomplexe Grössen und Darstellungstheorie in arithmetischer Aufassung*, i.e., Division algebras and representation theory from the arithmetic point of view.

127 See [Reid 1970], p. 188.

128 Cf. [Siegmund-Schultze 2016], pp. 60–62.

extended by the President of the Organizing Committee of the future Congress in Bologna included also Germany, a country which had not yet responded to the invitation to join the IRC. "This severe lapse", wrote the Secretary General of the IMU129 in his letter of 29 May 1928 to the President of the Executive Committee, "renders all these invitations illegal. . . . In view of the conditions under which these invitations have been issued it is no longer possible to consider the Bologna Congress as being held by the IMU. Hence, and after having conferred with the President of the IRC130, it is impossible for me to ofcially invite our members to the Congress that the University of Bologna will have organized under the presidency of its Rector."131

In response to this, Pincherle—who after all was also the President of the IMU wrote a letter to Picard132 explaining that the politics of exclusion was no longer acceptable to many colleagues abroad; he mentioned explicitly The Netherlands, Denmark, Sweden, the US, and the UK. He added that this was also the point of view of the Italian government, whose head Mussolini "grants the congress his moral and material support."

In the end, leading French mathematicians like Picard and the IMU Secretary General Kœnigs did not show up at the Congress, but Pincherle had made sure beforehand that others did come, for instance Élie Cartan.133 Altogether the French delegation counted 56 members.

In his letter to Picard, Pincherle had still expressed the hope that the IMU would hold a regular assembly at Bologna which could discuss the path to take for the future. But since Kœnigs did not cooperate, the gathering which Pincherle convened in Bologna would be considered as informal. Pincherle declared his resignation as President of the IMU on this occasion. In January 1929, William Henry Young whom we frst encountered in Section <IP_ADDRESS> above—took his place.

Young, his wife Grace Chisholm Young, and their daughter Cecily had participated in the Bologna ICM. He was the Vice-President of the Congress representing England; his wife acted as Vice-President of the broad Section I–B which spread from set theory to quasi-periodic functions. In the following years, Young would engage himself in ways that remind us of Fields's tireless trips (both men, by the way, were born in 1863), in order to keep the IMU alive.

He decided to make a personal trip to the countries not yet belonging to his Union to persuade them to join it; but on his part it was much more than a visit by the President of an international organization. He saw it as his trip down the map of Europe to meet its leaders and convince them of the importance of his work. He spent part of the summer of 1929 in London arranging not only his visas but also audiences with Kings and Prime Ministers of the countries that he would be visiting. He made his trip in two parts, and at his own expense. The frst lasted from September to December, and took in Poland, Austria,

129 This was Gabriel Kœnigs. Cf. [Capristo 2016] for more comments on the exchange with Kœnigs. 130 I.e., Émile Picard.

131 See Proceedings ICM 1928, Vol. I, pp. 7–8; my translation.

132 It is reproduced at length in a two-page footnote, Proceedings ICM 1928, Vol. I, pp. 8–9.

133 See [Capristo 2016], pp. 300–304. The article also discusses hypotheses about Émile Borel's absence proposed in [Bru 2003].

Hungary, Serbia, Roumania, Bulgaria, Turkey, Greece and Italy, while the second took place between April and June 1930, and included Germany, Denmark, Sweden, Finland, Latvia, Estonia and Czechoslovakia.

It was a great personal success for him: he saw ministers and leading mathematicians in all the countries visited, and even secured a long audience with the King in Bulgaria. He also gave lectures in English, French, German or Italian on the aims and ideals of his Union, and took the opportunity to learn something of the languages of countries visited with which he was previously unfamiliar.

A number of countries, especially in Eastern Europe, did in fact join the Union and the Council as a result of Will's eforts; but the main problem, which remained unsolved was, of course, the enrolment of the Germans.134

An apparently poorly prepared assembly during the 1932 Congress in Zürich would not manage to unwedge the IMU. An additional adverse efect may have been played by the fact that, according to the 'sunset clause', the Convention under which the IMU had been constituted would lapse on 31 December 1931, unless renewed.135

Thereafter IMU was *in limbo*, even if a few colleagues probably still hoped for a resurrection in 1936 which, however, would not take place.136 At the same time world politics brutally transformed international relations. For example, Hitler would withdraw Germany from the League of Nations in October 1933, but in the sequel the Nazis would consciously try to showcase selected German scientists at international meetings. Mussolini's Abyssinian war started in October 1935, however, would lead to a boycott of Italy issued by the League of Nations and observed in particular by Norway, with the consequence that the Italian government would not authorize Italian delegates to participate at the Oslo ICM in 1936.

Pincherle's realistic and adroit politics had brought about the frst ICM with more than 800 active participants, the biggest one to be assembled before 1950. Besides it could conveniently be claimed to be in certain ways 'apolitical', never mind that its coming about was a political feat *par excellence*. It was a resounding success and opened the gate for two other amazingly successful ICMs in 1932 (Zürich) and 1936 (Oslo). All this seemed to corroborate the old idea that mathematicians could very well organize ICMs without being framed by international institutions. This after all was what they had been doing since 1897.137 However, Bologna demonstrated even more: that, backed by the peculiar coalition of a Locarno spirit and the

134 See [Grattan-Guinness 1972], p. 175.

135 We have mentioned this with respect to the IRC in Section 4.1, shortly before Section 4.1.1. The corresponding clause in the IMU statutes was Article 18: *La présente Convention est valable depuis le 1er janvier 1920 jusqu'au 31 décembre 1931. Après cette date, elle sera renouvelée pour une autre période de douze ans, avec l'assentiment des pays adhérents.*

136 See [Lehto 1998], pp. 50–60. Cf. in particular Henri Fehr's attitude and the history of the ICMI, which we discuss in Chapter 9.

137 The fact that the ofcial title of the Bologna Proceedings ends with the line: "BOLOGNA 3–10 SETTEMBRE 1928 (VI)" has been occasionally read as a new count of the ICMs according to which Bologna would have been ICM number 6, so that the ICMs in Strasbourg and Toronto controlled by the IRC and the IMU, and without a German presence—were not to be counted; see for instance [Curbera 2009], p. 89. In fact, the Roman numeral VI added to the year 1928 indicates the 'sixth year of the fascist era' in Italy: 28 October 1927 – 27 October 1928. This sort

national interests of a fascist regime138, mathematicians from various nations could elude the rules of an International Council and its International Mathematical Union.

From its very conception in 1919—see Section 4.1.1 above—the frst IMU had quite a narrow scope which never went substantially beyond ideas for schemes of reviewing the mathematical literature (which never grew to fruition)139 and holding International Congresses.140 This has provoked understandable criticism, for instance from Olli Lehto in his history of the IMU: "It is striking how few scientifc activities the Union undertook. This lack of mathematical substance was a serious faw. It played a role in the decline of the Union, which became increasingly obvious from 1928 on."141 For instance, there was no IMU activity resembling the Encyclopedia project with its strong international aspect (the Encyclopedia itself—see Section 1.3.3 above—was based in Germany, and thus of course anathema for the IMU). Yet, at the Fourth General Assembly of the IRC, on 13 July 1928, the ICM Proceedings were generously acknowledged by Émile Picard as a kind of IMU equivalent to international publication projects of other scientifc unions:

Gentlemen, this is our fourth general assembly and the edifce we have erected in 1919 is now in the ninth year of its existence. I do not know what the future reserves for us, but it is certain that the creation of the IRC has been extremely useful. While before 1914 there were a lot of groupings whose tasks often overlapped, but which where unrelated to one another, the Council has succeeded in catalyzing the creation of a rather limited number of Unions, each dedicated to a corresponding branch of science. They work in the same spirit, and albeit obeying certain general rules—enjoy great freedom of action. Their eforts, at least for some of them, have been very fruitful. It is with legitimate satisfaction that we are reading the important volumes published by the Geodesic and Geophysical Union, the Astronomical Union, the Chemical Union, and others. And even Unions dedicated to sciences which are much less prone to collective work have published highly interesting studies, such as the International Union of Mathematicians in its meetings at Strasbourg and Toronto.142

The Bologna Congress opened just three weeks after Picard's speech. The six big volumes of its Proceedings would no longer be claimed in the name of the IMU or the IRC. They mention the Council and the Union only as annoying political hurdles during the preparation of the Congress.

of calendrical political alignment had been mandatory in Italy since 1927. Personally, I am not aware of any public attempt to play with the numbering of ICMs before Hermann Weyl's exuberant speech at the Zürich ICM in 1932—see Proceedings ICM 1932, Vol. I, p. 71.

138 The crucial importance of the fascist backing is highlighted by the fact that Pincherle thanked Mussolini for his help and informed him of the success of the Congress in a letter dated 11 September 1928, i.e., the very frst day after the closure of the event. See [Giacardi & Tazzioli 2019], pp. 40–41; cf. the discussion in [Giacardi & Tazzioli 2021] of the echo of the Congress and its fascist connections in the *Bolletino* of the *Unione Matematica Italiana*.

139 Cf. Section 6.3 below.

140 The International Commission on Mathematical Instruction ICMI—whose creation in 1908 we have briefy mentioned in Section <IP_ADDRESS>—played a slightly special role though. Cf. Chapter 9 below.

141 See [Lehto 1998], p. 33.

142 See [Schuster 1930], p. 2; my translation.

Based in part on his unsuccessful search for documents relating to the IMU in the IRC archives, Lehto concludes his refection by writing: "In all, the old IMU had poor visibility within the International Research Council and was not well known among mathematicians."143 In contrast to this, I have tried to show that both the IRC and the frst IMU were in fact widely known among mathematicians in the 1920s, on both sides of the divide brought about by World War I. However, their reputation was profoundly and indelibly political, rather than scientifc.

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143 See [Lehto 1998], p. 33.

## **Chapter 5 Philanthropic Capital for Mathematics**

The US-based activities that we briefy recall in this chapter are an important reminder, within the crystalline sphere of scientifc endeavors, of the truly global transformation that World War I had wrought:

In November 1918 Germany's planned economy surrendered in the face of a second even more powerful economic vision—a triumphant model of 'democratic capitalism.' At the heart of the democratic war efort stood the much-heralded economic potential of the United States. World War I marked the point at which America's wealth stamped itself dramatically on European history.1

After the war, American philanthropy was often able to provide what European states could no longer aford. This held true for countries on both sides of the war frontlines:

I expected to fnd some outstanding diferences between the victors and the vanquished in the late war, at least in so far as economic state, after-war national psychology, etc. might afect the higher education in these countries; in this respect I was quite wrong for some of the victors seem to be in quite as bad a state as any of the vanquished.2

The support of scientifc projects was realized and acted out according to the principles of American philanthropy and guided by the US scientifc perspective and expertise. Indeed, American philanthropy had begun its tremendous works of donation well before the frst World War. It had already marked its durable imprint on the academic landscape of the US in the nineteenth century. In Chapter 1 we had several occasions, for instance, to mention the university at Baltimore, which had been endowed by Johns Hopkins. What interests us here, however, is how such large scale private donations to science went international after World War I.

As far as mathematics is concerned, rich new sources of international support began to fow in the mid-twenties, i.e., at about the same time as the relevance of the IMU for Mathematics International started to dwindle. Focussing on research training at the highest level they single-handedly set a new standard for the intercontinental framing of scientifc excellence. Scientifc ideals which had been built up and

1 See [Tooze 2014], p. 200.

2 Augustus Trowbridge as quoted in [Siegmund-Schultze 2001], p. 56.

cultivated in the USA, largely with European examples in mind, now made a strong reappearance in the applications for funding submitted by Europeans. As humanity moved closer to World War II these philanthropic resources were malleable enough to be increasingly used to relocate refugee scientists in America.

#### **5.1 The Rockefeller Philanthropies**

In the context of the present book it is legitimate to focus on visible, international efects of US philanthropy for mathematics. It is also requisite, since no history of mathematics of the 1920s and thirties would be adequate if it failed to record the impact of fnancial support on international mathematical networking, especially that provided by the Rockefeller Foundation. The only systematic study to date of the Rockefeller Foundation's activities in the domain of mathematics is the book [Siegmund-Schultze 2001].

The vantage point of the history of mathematics does not project a fair image of American philanthropy in general,3 nor does it duly capture the global scale of an organization like the Rockefeller Foundation—mathematics was lagging behind in globalization. Already *The Digital History* ofered today on the Foundation's website gives a frst impression of the true breadth of the activities at the time. Activities to improve health care, both in America and on other continents, stand out, but archaeology, literature and theatre also show up. The activities for mathematics, which did not have its own explicit subheading, fall into the category of *Natural Sciences*.4

Yet there was one period in the history of the Rockefeller Foundation when mathematics was treated in practice as if a dedicated line of expenditure for it existed. This was when the grants were handled by the Foundation's *International Education Board* (IEB), created in 1923. The time spell ended in 1931. Afterwards the fnancing of mathematics went over to the Rockefeller Foundation, which survived, unlike the IEB, but accepted only a relatively small number of mathematicians. Even so, among these later grantees one fnds well-known mathematicians of the twentieth century, for instance the winner of one of the frst two Fields Medals in 1936, Lars V. Ahlfors (1907–1996), the Polish logician Alfred Tarski, and the British algebraic geometer John Arthur Todd (1908–1994). On the other hand, applications in the 1930s by outstanding mathematicians such as Andrey Nikolaevich Kolmogorov (1903–1987) or re-applications by former IEB fellows like Stefan Banach (1892–1945) and Bartel Leendert Van der Waerden were dismissed, sometimes ofcially on the grounds that there was no specifc program for this feld of knowledge. On the other hand, as the

3 For a general, political history of American philanthropy, see for example [Zunz 2012].

4 See [URL 07]. Even though there is no special section dedicated to mathematics, browsing the site one does fnd a few related documents, for instance about John von Neumann, or on Vannevar Bush's *Diferential Analyzer*.

1930s wore on, some mathematicians could proft from Rockefeller grants given to political immigrant scientists in the US.

Let us introduce the IEB in general by quoting from a portrait penned in 1941 by way of a summary of the book [Gray 1941]:

The International Education Board was set up in 1923. The idea behind it was to help to make good some of the ravages of the War of 1914–18. The money, which amounted in all to nearly twenty-eight million dollars, was provided by John D. Rockefeller, jun., who . . . imposed no conditions on the manner in which it should be spent, except that it should be used for 'the promotion and advancement of education throughout the world.' The inspiration with regard to the policy which should be followed came almost entirely from . . . Dr. Wicklife Rose [1862–1931]5.

What is education? In Rose's mind it became for the most part, not the dissemination of certain accepted ideas and cultural patterns, for that he felt might well be left to the various national Governments, but the desire to forward the understanding of the natural world by the best possible means. The claims of educational training, particularly training for agriculture, were not overlooked, but they played a subsidiary part in the comprehensive scheme which he put forward for the support of the best research institutions and the most promising scientifc workers, whose work was being held up for lack of funds. . . . 6

In other words, at least for the natural sciences, the meaning of *education* in the name of the IEB was narrowed down to the most advanced sense, i.e., education towards high-level research. The funding for institutes was conceived accordingly as providing solid structures for research training. In particular, despite its name, the International *Education* Board did not till the same soil as the ICMI did in the feld of mathematics—cf. Chapter 9 below.

No considerations of national prestige were allowed to stand in the way, and except for agriculture, no attempt was made to strike a balance between the competing claims of the diferent branches of science, for in Rose's view, 'all knowledge is inter-related, and if we help in any one feld we help in all the others.' So it came about that the greatest scene of the Board's activities lay in Europe, including the British Isles; but a small number of individual projects in the United States received some of the largest grants, while smaller ones found their way to such places as South Africa, China, the Philippines and New Zealand.

In all, 'ffty-seven universities, research centers, and other institutions were provided with new buildings, equipment, endowment and other material aids; and 603 individuals, chosen for their promise of future usefulness, were assisted in their higher education, given opportunity to study under world authorities in their chosen felds, introduced to new pastures of research under conditions which at the time seemed favourable to their development. Through grants for these various purposes, thirty-nine countries, representing Europe, Africa, Asia, Australasia, and the Americas, were aided.'7

Thus the IEB aid was essentially spent on two diferent kinds of projects: personal stipends and the funding of constructions for outstanding research centers. The question arises how IEB went about choosing the persons and institutions to be supported.

5 On W. Rose, cf. [Siegmund-Schultze 2001], pp. 27–30.

6 See [Weatherwall 1941], p. 398.

7 See [Weatherwall 1941], p. 398.

A visit of Dr. Rose to Europe in 1923 initiated a scheme under which the whole world, but particularly war-worn Europe, was scoured for young scientifc workers showing exceptional promise, whose studies were held up through lack of means. After careful scrutiny these were granted travelling fellowships for a year, which enabled them to proft by the best scientifc experience available in the world in their own particular line. Within the fve years, 1923–28, an exchange of workers and of scientifc ideas took place on an unprecedented scale.

But this scheme of fellowships in science would have been held up by the cramped facilities existing in many of the leading research institutions. Realizing this, the International Education Board made available large sums to be spent upon buildings, equipment and endowment. One of the frst institutions to beneft in this way was the Institute of Theoretical Physics at Copenhagen, under Niels Bohr [1885–1962]. . . . 8

Starting with Rose's European journey, the IEB began—in the domain which interests us here—to design a map of mathematical Europe, or rather, of the Europe of mathematics and physics; of major persons and centers in the principal countries.

Both for the UK and for the overall constellation of European mathematics, Godfrey Harold Hardy was a key person to talk to.

The leading English mathematician Godfrey H. Hardy had been one of the frst European scientists to be contacted by Rose during his trip to Europe in the fall of 1923. The two men met in Oxford, on December 23, 1923, and Rose got advice about promising mathematicians in Europe but no request proper from the English side. This changed when Rose got back to England, shortly before leaving Europe, and met Hardy in London, once again, on April 14, 1924.9

Hardy's role as correspondent for the IEB goes well beyond his own research felds, analysis and analytic number theory; it fts in with his outspokenness in favor of Mathematics International since World War I. Already during the war Hardy sternly refused to transport national preferences into scientifc life. For instance, the Latin dedication of the joint book [Hardy & Riesz 1915]—the fnal manuscript had to be fnished by Hardy himself; correspondence with his Hungarian coauthor was increasingly difcult—translates:

To the mathematicians (how many and wherever they may be): that they may soon again take up, as is to be hoped, the confraternity of their works which is currently disrupted, we, the authors, friends and foes at the same time, present and dedicate [this book].10

The war years put him at odds with most Cambridge colleagues. In 1919 he accepted the Savilian chair in Oxford. This is where Rose frst met him to hear his views on mathematical Europe. It was also from Oxford that Hardy intervened in or commented on many of the correspondences about the exclusion policy of the IRC,

8 See [Weatherwall 1941], p. 398. This article was written during World War II; on p. 401 one reads: "some of the work of the Board is already in ruins." The piece ends on a disillusioned note, timidly hoping for a brighter future after the war.

9 See [Siegmund-Schultze 2001], p. 40; cf. pp. 247–249 for Hardy's note addressed to IEB at the second meeting, on behalf of the London Mathematical Society.

10 Cf. [Corry & Schappacher 2010], p. 435. For Hardy's own account of World War I in Cambridge, see [Hardy 1942]. Cf. June Barrow-Green's chapter in [Aubin & Goldstein 2014], pp. 59–124.

the IMU, and the frst post-war ICMs which we have mentioned in the preceding Chapter 4.

As of 1925 an ofce in Paris was established under the direction of the rich polyglot Augustus Trowbridge from Brooklyn, New York, who had obtained his PhD in Physics at the University of Berlin in 1897. His regularly kept diary ("log") is one of the central sources exploited by Siegmund-Schultze for the book [Siegmund-Schultze 2001]. Trowbridge was well-connected with the Paris scientifc milieu. And for situating the merits and needs of European mathematics at large he could rely on reconnaissance missions undertaken by the leading American mathematician George David Birkhof (1884–1944), the father of Garrett Birkhof (1911–1996).

. . . George David Birkhof travelled to Europe together with his family in the second semester of the academic year 1925/26 (probably starting in February 1926). He had planned a shorter stay in Europe within a sabbatical year, but stayed several months longer (until September 1926) on the basis of the support given by the IEB. Birkhof chose France as his temporary home and country of departure for various trips to several European countries. In Paris he collaborated closely with American physicist Augustus Trowbridge, who was heading the European ofce of the IEB in the city. At the end of his journey, on 8 September 1926, Birkhof submitted to Trowbridge a 12-page-long "Final General Memorandum for Dr. A. Trowbridge."11

In terms of physical constructions, these explorations resulted in two new buildings for mathematics, both granted by IEB on the same day in December 1926, and both inaugurated in 1928: the *Institut Henri Poincaré* (IHP) in Paris, and the *Mathematisches Institut* of Göttingen University.12 Each one of them was apparently seen by the American donors as a contribution to a scientifc campus. This vision ft reasonably well with the pre-existing buildings for physics, fuid mechanics, and chemistry in Göttingen near which the new Mathematical Institute was built. Also the *Institut Henri Poincaré* found itself close—in fact, very close—to other institute buildings (like chemistry and oceanography) that had recently been fnished. This condensed Pierre and Marie Curie 'campus' is in the vicinity of Sorbonne University from the turn of the century, which is architecturally much more confned and squarely occupies full city blocks with internal courts. Lecture halls, a rich library, reading and seminar rooms, and also collections of mathematical models, were the visible assets for research training in both new institutes fnanced by the IEB.

Already in the Summer of 1924, Gösta Mittag-Lefer's application for IEB funds to insure the survival of the *Institut mathématique Mittag-Lefer* had been turned down. He and his wife had decided to set up a foundation around the extraordinary mathematical library in the generous villa at Djursholm, outside of Stockholm,

11 See [Siegmund-Schultze 2001], p. 46; Birkhof's memorandum is reproduced there on pp. 265– 271.

12 See Chapter V of [Siegmund-Schultze 2001]. I have also greatly profted from an inspiring lecture comparing both buildings from the point of view of the history of architecture, delivered by Bernd Hofmann, Göttingen, at the eightieth birthday celebration of the IHP in 2008.

**Fig. 5.1** Rockefeller map of mathematical centers in Europe, 1927. The colours indicate the various branches of mathematics: analysis, geometry, applied mathematics, theory of numbers, algebra, philosophy. Credit: [Arch. RAC].

Sweden. The *Institut mathématique Mittag-Lefer* had been formally created in 1919. The main reason for the IEB to abstain from helping this splendid site was apparently its isolated situation.13

The denial to fund Djursholm in contrast highlights the IEB's expectations to the efect that the grants for the IHP and Göttingen would create attractive and lively research centers. Paris was of course well chosen in this respect already because of the city's attractiveness for students, also from abroad; in fact, Rockefeller money also went into the construction of the central building of the Paris *Cité universitaire*, a complex of international student residences in the South of the capital. Mathematically, the newly founded IHP would play a particularly visible role in the development of mathematical statistics and probability theory.14 As to Göttingen, the proximity between mathematics and physics seems to have played an important part in convincing the IEB to invest in this place. In Section 6.1 below about Emmy Noether's legacy, we will analyze the peculiar purely mathematical message which young researchers would pick up there and spread in the 1920s and early 1930s.

13 See [Siegmund-Schultze 2001], pp. 178–180.

14 See [Catellier & Mazliak 2012], and [Siegmund-Schultze 2001], pp. 169–175.

As to the individual grants extended to mathematicians, between 1924 and 1931, the IEB fnanced research sojourns of a total of 86 predominantly young mathematicians; three women and 83 men. Their felds of interests ranged widely, from logic, via all principal domains of pure mathematics, to applied felds like aerodynamics and statistics. Since Rockefeller grants for mathematicians became rather the exception after 1931, only the IEB period represents a fair measure of the internationalizing efect of Rockefeller money for mathematics. In terms of nationalities, 16 Germans, 14 Americans, among them 2 women, 11 Frenchmen, 6 men from Poland, 5 men and one woman from the USSR, 4 Austrians, 4 Czechs, 4 Hungarians, 4 Swiss; 3 men each from Holland, and the UK; 2 men each from Italy, Norway, Romania, and the Kingdom of Yugoslavia; and one mathematician each from Finland, Greece, and Japan received IEB grants.15 So we are looking essentially at an afair between Europe and the US. Since they are so strongly represented, let us take a quick look at the IEB grant recipients from Poland and the USSR.

The Polish mathematicians who received IEB grants were the emblematic Stefan Banach, the topologist Witold Hurewicz (1904–1956), the analyst Szolem Mandelbrojt (1899–1983)—who had actually been based in France since 1920, would obtain French citizenship in 1927, and become Hadamard's successor at *Collège de France* in 1938—, the famous statistician Jerzy Neyman (1894–1981), the expert in fuid mechanics (and diplomat in his later years) Piotr Szymański (1900–1965)16, and the analyst Antoni Zygmund (1900–1992). All of them went to Paris at least for part of their grant, except Hurewicz who spent the academic year 1927/28 in Amsterdam, hosted by L.E.J. Brouwer.

The Russian topologist Pavel Alexandrov (1896–1982) also spent a year (1925) in Holland, welcomed by Brouwer,17 and in 1826–27 he was granted 8 months in Princeton,18 invited by Solomon Lefschetz (1884–1972). Alexandrov's former teacher Luzin, in spite of the fact that he could no longer claim to be a young researcher, was fnally granted a stay in Paris in 1928, one year after his former student Dmitrii Menshov (1892–1988). While Menshov was recommended by Arnaud Denjoy (1884–1974) and Paul Montel (1876–1975)—as well as by his teacher Luzin, Luzin was backed by Lebesgue; yet he had to try twice before he was admitted. In his French application, which is apparently difcult to translate, Luzin concentrates on set theory, adopting the point of view of *naming infnity*, which we have briefy touched upon in Section 2.1.1 above.19 The same year Luzin also participated at the Bologna ICM where he sketched his take on the foundational debate.20

15 See [Siegmund-Schultze 2001], pp. 288–301, for the total list of 130 mathematicians known to have either received IEB grants or to have been sponsored by Rockefeller grants later in the 1930s; see pp. 96–106 for remarks on the lucky and some of the less lucky applicants.

16 See [Urbanowicz & Tijsseling 2016].

17 For more background on this stay, in particular Pavel Urysohn's (1898–1924) work and tragic death and Emmy Noether's role, see [Rowe 2021], pp. 109–120.

18 This is Princeton University. The *Institute for Advanced Study* did not exist yet—see Section 5.2 below.

19 See [Siegmund-Schultze 2001], p. 250. Cf. [Graham & Kantor 2009], esp. pp. 205–211.

20 See Proceedings ICM 1928, Vol. 1, pp. 295–299.

The only non-American woman among the IEB fellows, Nina Karlovna Bari (1901–1961)21 had also been a student of Luzin's. She had profted from the opening of the universities for women in 1918 as a consequence of the Bolshevik Revolution and was in fact the very frst woman to graduate from Moscow State University. Nina Bari would become full professor there in 1932. At the Bologna ICM in 1928 she presented in a sectional talk the peculiar result to the efect that every continuous function on a real interval is the sum of at most three functions of the form ◦ , with both and absolutely continuous. Thanks to the IEB, Nina Bari could spend nine months in Paris in 1929.

The remaining two IEB fellows from the USSR were: Abram Samoilovitch Besicovitch (1891–1970), who used his IEB fellowship as a stepping stone towards his future career in the UK, and the complex function theorist Vasilii Leonidovitch Gontcharov (1896–1955), who would later be known in the USSR for his elementary textbooks.22

**Fig. 5.2** Rockefeller map of mathematical centers in the USA, 1927. The colours indicate the various branches of mathematics: analysis, geometry, applied mathematics, theory of numbers, algebra, philosophy. Credit: [Arch. RAC].

21 Both in the Italian Bologna Proceedings and in many of her publications in Western media, her name is transliterated as 'Bary', even though her Russian name ends with a single plain letter 'i.' 22 Cf. [Siegmund-Schultze 2001], pp. 125–132, for a general discussion of IEB's perspective on Soviet Russia.

Before closing this subsection, we ought to remind ourselves that the Rockefeller Foundation was only the biggest actor in a broad feld of philanthropic initiatives. This is obvious in the realm of mathematics when one looks at fellows of the *John Simon Guggenheim Foundation*, even though these grants, which started roughly at the same time as those of the Rockefeller Foundation, were limited to US citizens or residents. Their list for the second decade, 1935–1945, includes some of the most infuential mathematicians of the twentieth century, such as Paul Erdős (1913–1996), Marshall Harvey Stone (1903–1989)—who would play a dominant role in recreating the IMU after World War II—, as well as the rewriters of algebraic geometry Oscar Zariski and André Weil.23 Many of these later grants concern mathematicians who had emigrated to the US before. This will be put into perspective in Chapter 7 below. Actions of the Rockefeller Foundation in the 1930s will also be recalled coincidentally in Section 6.1.

For the time being, let us summarize for the record that the activities of the IEB and similar philanthropic foundations in the domain of mathematics led to the construction of two mathematical institutes, in Paris and in Göttingen, and provided generous help to a fair number of researchers. Since these actions were embedded in a very broad range of philanthropic activities, sweeping the spectrum of scientifc domains and sometimes encompassing geographic regions which were not on the mathematical map yet, one may say that philanthropic convictions and a US-based analysis of the scientifc world did more to promote a certain spirit of Science International between the two World Wars than any other international institution. Furthermore, they were major novel steps in pushing scientifc practice towards the constant collegial interaction beyond local contacts that we take for granted today.

Indeed, foreign travel has been a recurring theme in our survey of the nineteenth century world of mathematics—see Chapter 1. For well established or wealthy actors such as Guccia, Klein or Mittag-Lefer it was a natural part of their networking strategies. Young researchers on the other hand like Betti, Brioschi, Casorati (Section <IP_ADDRESS>) undertook their journey in order to discover the world of mathematics. Sofya Kovalevskaya had to leave her home country in order to study in the frst place, and then again to embark on her academic career—see Section 1.1.7. But all those journeys were private initiatives. Typically, at least for the younger participants, participating in international congresses, with the resulting contacts and exchanges, had to be arranged privately. The idea of helping promising young mathematicians by systematically granting them the opportunity to spend time at a suitable institution only gained ground between the World Wars, as a relatively late ingredient of the ongoing professionalization of science. On the national levels it typically expressed itself through newly founded National Research Councils. The international dimension, at frst covering mostly Europe and the US, was opened up by American philanthropic initiatives.

23 See the complete list of Guggenheim fellows in mathematics up until 1945 in [Siegmund-Schultze 2001], pp. 302–303; see also pp. 138–139.

#### **5.2 The Institute for Advanced Study, Princeton**

Like the activities of the IEB, the founding of the Institute for Advanced Study (IAS) at Princeton in 1930 was also the result of American capitalist philanthropy. Here too a model institution for worldwide science came into being.

[T]he Institute holds a special symbolism for mathematicians. The Institute for Advanced Study began in 1930 through the vision of Abraham Flexner [1866–1959]. Flexner was a fgure of considerable infuence during the frst half of the twentieth century. He made his mark in 1910 with a scathing exposé of the defciencies in American medical education. Flexner's revelations called for drastic action. Over a decade-long period he served as the architect of a Rockefeller philanthropic initiative that dramatically upgraded American medical schools.

When Flexner retired from the Rockefeller Foundation, it was with the satisfaction that his career had been essential to the modernization of American medicine. Still, he had a distinctly diferent ambition that remained unfulflled. As a long-time observer of higher education, Flexner was convinced that the United States should possess an exclusively graduate university with an ideal environment for research. There, a small faculty of geniuses would direct the studies of a few disciples while pursuing their own discoveries.

With the power to direct millions of dollars to selected universities and hospitals, Flexner had accumulated a stunning collection of contacts among academic, business, medical, and political leaders. When department store magnate Louis Bamberger [1855–1944] and his sister Carrie Fuld [1864–1944] began seeking advice on devoting their fortune to the creation of a new medical school, it was inevitable that their consultations would lead them to Abraham Flexner. Out of these discussions Bamberger and Fuld decided to endow a graduate university with the 63-year-old Flexner as director.24

Flexner worked hard—trips to Europe included—to recruit outstanding scientists with international visibility on the faculties of the frst 'Schools' he was planning for the Institute. Even though Flexner was personally as clueless about mathematics as Alexander von Humboldt a century before him, he set his mind on building a School of Mathematics. Solomon Lefschetz from Princeton University recommended to go after "the younger group of geometers. It is the most vital and promising of mathematical groups in the U.S., the one with the highest national and international standing. It includes [Oswald] Veblen [1880–1960] and [James W.] Alexander [1888–1971] of Princeton, [G.D.] Birkhof and [Marston] Morse [1892–1977] of Harvard and also myself." Furthermore, Lefschetz remarked: "Hermann Weyl is the only mathematician anywhere defnitely above these names. But as he occupies the most distinguished mathematical chair in the world (in Göttingen) I do not see him giving it up."25 Whereas Birkhof decided to stay at Harvard, Veblen joined the new institution in 1932, and the topologist Alexander followed in 1933. John von Neumann (1903–1957) was also hired in 1933. Morse would join in 1935. In terms of international luminaries, Flexner landed a brilliant success with Albert Einstein who arrived in 1933. It was Adolf Hitler's regime which fnally decided the hesitating

24 See [Batterson 2006], p. ix.

25 See [Batterson 2006], p. 59, for these quotes from Lefschetz.

Weyl to leave Göttingen for Princeton in the Fall of 1933; the Nazi rule not only clashed with Weyl's democratic convictions after the 17 years he had spent in Zürich, it also potentially threatened his Jewish wife, and children.26

Thus within a few years, this new institute was one of the strongest mathematical centers in the world, and we shall see in later chapters how prominent a role it would play on the international scene of mathematics. But it is not so much the individual IAS which interests us, and how it continued to hold its eminent place for global mathematics. If we discuss its founding here, it is because this was in fact the birth of a new type of research structure, which would subsequently serve as a blueprint for similar centers founded all around the globe since the end of World War II. So what was this institute like, once its structure had crystallized? How can the IAS, in particular its School of Mathematics, be described once it was set up and functioning; for instance, when it had moved from its frst, provisional quarters in Princeton University, to Fuld Hall in 1939? The original idea of a graduate school never materialized, and was apparently abandoned early on: "From the beginning the faculty identifed and hosted scholars who had already received their doctoral degrees. These visitors, who became known as members, typically remained at the Institute for a year."27

Here is a concise description of this new kind of research site whose very frst example was the IAS's School of Mathematics: It is a (relatively) independent academic structure (even though it may have collaborative ties with local academic institutions, for instance universities, nearby). Its goal is to support fundamental research in mathematics (and possibly also other domains) at the highest level of intellectual inquiry. A rather small permanent faculty selected for their outstanding research record guides the work, and each year up to about seven times as many visiting members are invited to join the Institute, from universities and research institutions throughout the world. Every researcher, permanent or visiting, is free to pursue their personal research agenda. Seminars are organized, talks are given, which refect ongoing work. In addition, the institute ofers numerous informal occasions for exchange.28

We will call an institute that fts this mold a *Locally-grounded Transnational Research Site for mathematics*, or LGTRS for short. This is our adaption, specifically for the domain of mathematics, of a notion that historians-sociologists of (predominantly experimental) science have coined to capture important elements of the professionalization of scientifc research in the second half of the twentieth century:

Before World War I, the threat of cognitive fragmentation came from the massive and rapid introduction of additional specialties. Today this menace is linked to the existence of several representation systems inside almost every major scientifc discipline. Within

26 For many more details packed in an entertaining narrative, cf. [Batterson 2006].

27 See [Batterson 2006], p. x.

28 Some of the formulations of this paragraph are slightly adapted clippings from the website [URL 08]. The approximate factor of 7 linking the number of permanent faculty and invited members corresponds to the current situation at the IAS. It is cited here as an indication only; I will not just use any precise value in order to rule out a potential LGTRS.

a given feld, scientists choose between numerous alternative ways of representing their phenomena. . . . Some groups emerge within disciplines whose principal loyalty lies with their chosen representation rather than the feld. This dramatically afects the pattern of work since scientists sharing a representation system often forge more meaningful and stable intellectual relations with colleagues of the same representation than they do with their specialty or home-laboratory. Moreover, efective use of a new representation to solve a particularly thorny problem, or strategies to gain legitimacy for a novel representation system spur scientists to band together for a period as they shift for a short while their research to a new site. . . . Today, scientists have ample opportunities to seize advantages lying outside their laboratories without the need to transfer to a new laboratory, agency or nation. Most research agencies provide funding and sometimes encouragement for short stays in an alternative laboratory. . . .

In the 19th century scientifc travel was a form of ambassadorship or assumed the form of brief laboratory visits intended as fact-fnding missions. In the 20th century, however, scientists travel either to cooperate with colleagues or to carry out gritty research that is better done with resources located away from their customary base of operation. The logic of LGTRS is hence thoroughly functional.

This leads to the emergence of new ties. Scientists' bonds with their institutional base is supplemented by an additional network consisting of individuals and laboratories scattered around the globe. In some instances, involvement with transnational groups, projects, and institutions becomes overriding, thereby neutralizing afliation with the home-setting. Here, the local/national coordinate system is countered by the appeal of LGTRS. . . .

Yet, to portray the relations between LGTRS and nation-based research as antagonistic would be to misunderstand this recent and crucial phenomenon. LGTRS are not a professional, cognitive or educational alternative to national science. They constitute an incremental resource as scientists attempt to expand and multiply strategies and techniques for problemsolving. The LGTRS dovetail the local, regional and national endeavors. Scientists operate simultaneously and on the three planes in complete comfort and without the slightest sense of contradiction or alienation. The salient feature of this new aspect of research practice and organization is oscillatory movement of individuals away from and going back to their home-base. Centrifugal and centripetal trajectories succeed one another as required by the research projects.29

This analysis clearly takes into account the importance of experimental devices such as a Hadron Collider, a supermagnet or the like. It nonetheless also describes very well the crucial changes that have afected professional mathematical research as of the middle of the twentieth century. These novel features, which were frst realized at the Institute for Advanced Study and may today seem banal (at least in the countries that are fully integrated into the world mathematical community), were intrinsically international.

The ambience in Princeton, which is still fairly cosmopolitan, was even more so in 1937. The Institute for Advanced Study did not yet have its own buildings; the University provided it with comfortable facilities in the old Fine Hall, to which Veblen had devoted so much care, but guests such as I were left to their own devices as far as housing went. Such stays are fruitful, but the experience has become such a common one that any remarks I could make would be superfuous. As planned, I gave a series of lectures on the topic of my future

29 See [Crawford et al. 1992], pp. 28–30.

paper in the Journal de Liouville30, and it was no small boost to my ego to see Hermann Weyl among those who attended regularly. Through contact with Alexander, I tried to fnd out more about "combinatorial topology". . . 31

Flexner's original plan of a graduate school of mathematics for the IAS was fairly close to the kind of institutes that the Rockefeller Foundation helped building through its *International Education Board*. Already in this respect, and in spite of the narrow, elitist interpretation of the word *education* upheld by the IEB, neither the Rockefeller institutes nor the initial layout for the IAS were projects in the style of an LGTRS. The Göttingen Mathematical Institute fnanced by the IEB was an integral part of Göttingen University and actually contained a generous class room for graphical methods (*Zeichensaal*) open to students of all levels.

The *Institut Henri Poincaré*, on the other hand, in some ways resembled an LGTRS. In spite of initial difculties in view of the economic situation, nine chairs were fnally integrated into the plan for the IHP.32 While graduate teaching did take place in its lecture halls, the most visible and novel roles of the IHP in the interwar years was to host regular seminars, and to welcome mathematicians from many diferent countries, albeit usually for one or several lectures rather than for a prolonged stay.33

We will pick up the global history of the LGTRS model after World War II in Section 8.3 below.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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30 This alludes to the paper *Généralisation des fonctions abéliennes*, labeled [1938a] in [Weil 1980].

31 See [Weil 1992], p. 117.

32 See [Siegmund-Schultze 2001], pp. 157–168.

33 See [Siegmund-Schultze 2001], pp. 168, as well as the list of international lecturers at IHP on probability and statistics in [Siegmund-Schultze 2001], pp. 173.

## **Chapter 6 Mathematical Consolidation and Unifcation in the 1930s**

The 1930s may well be the most difcult decade of the twentieth century to come to terms with. Given what humanity was about to be led into, i.e. World War II, one may be tempted to consider it a "morbid age":

In his recent memoirs, the historian Eric Hobsbawm [1917–2012] remarked of the 1930s that 'we lived in a time of crisis.' Nothing very surprising about that. But I recall a conversation with him a few years ago, shortly before starting the research for this book, when he told me that he could remember a day in Cambridge in early 1939 when he and some friends discussed their sudden realization that very soon they might all of them be dead. This did strike me as surprising, and it runs against the drift of the memoirs, in which he argued that communists were less infected by pessimism than everyone else because of their confdence in the future. It is also very diferent from my own memories of life in Cambridge thirty years later in the late 1960s where, despite labouring under the shadow of the bomb and the threat of war in Europe during the second Czech crisis, students did not contemplate early extinction but preferred to listen to Leonard Cohen in rooms made mellow by too much smoke and cheap wine.1

For us who know how history went on, World War II looms over the thirties, but taking the decade from its very beginning the picture is hardly brighter. The great depression was sacking its ransom of losses and unemployed in most countries around the globe. A remarkably candid refection on the state of the world is contained in the farewell address to the mathematicians gathered for the closing session of the ICM (11 September 1932), pronounced by the mayor (*Stadtpräsident*) of the city of Zürich, Ständerat Dr. Emil Klöti (1877–1963):

Our city has done its best to present itself to our honourable guests in beautiful September sunshine so that we may hope it has won your respect, not only because it is so expensive, but also for the beautiful landscape. . . . Looking at the city and its surroundings you may have thought that it must be home to a happy smallish people.

But if you had had the occasion for closer contact with the inhabitants, it would have likely shattered this idea. True, Zürich is neither ugly nor poor. It even counts among the richest cities of the world, in which every inhabitant has a respectable average income. But you as mathematicians know what it means to take an average. . . . [A]t least we may say that the standard of living of the lower classes is quite a bit higher here than in many other places.

1 See [Overy 2009], p. xiii.

. . . The crisis which had long spared our country is now spreading every day. The number of unemployed is growing . . . and we look to the coming winter with apprehension. . . . Thus also in our population you will fnd right now more unrest and discontent than placid happiness.

Unfortunately the isolation of the individual states is not limited to the economic domain. It also encroaches on the people's mindsets. In these difcult times it is especially valuable if the various sciences cultivate and strengthen their international character . . . and maintain collegial contacts . . . regardless of race and nationality.

I hope, and I am confdent, that this . . . Congress has lived up to the task and contributed to upholding the international spirit and to defending it against those movements whose aim it is to pervert the natural love for one's homeland into unabashed nationalism and chauvinism.2

This speech was given less than fve months before Hitler became chancellor of Germany. And it was given less than six months before Frances Perkins (1880–1965) was sworn in as Secretary of Labor, the frst woman ever to serve in a presidential US cabinet. She would remain in this position for 12 years, through June 1945, proposing and implementing essential elements of President Franklin D. Roosevelt's (1882–1945) *New Deal*.

Among the younger participants at the 1932 ICM in Zürich were Henri Cartan, Jean Delsarte (1903–1968) and Szolem Mandelbrojt—who each gave a sectional talk—, Claude Chevalley (1909–1984)—who did not give a talk himself, but both Emmy Noether in her plenary lecture and Helmut Hasse (1898–1979) from Marburg, Germany, alluded to his work3—as well as Jean Dieudonné (1906–1992) and André Weil—whose results were mentioned in plenary lectures, by Gaston Julia (1893– 1978) and Francesco Severi, respectively. These six young men from France were the core of the group that would choose the collective pen name *Nicolas Bourbaki* for their joint project in 1935.

The way in which the Bourbaki project grew way beyond the analysis textbook initially intended, launching a comprehensive rewriting of mathematics as a whole, is an unmistakable symptom of 'Consolidation and Unifcation' which indicates that the overall image of the 1930s as a crisis ridden decade cannot be the whole story for the historian of mathematics. The objective of the current chapter is to try to put these symptoms of mathematical 'Consolidation and Unifcation' during the thirties into the context of what had happened in the previous decades, stressing both continuities and those novelties which set the 1930s apart. While doing this we will highlight the international dimension of the new decade, in spite of the IMU being essentially absent.

2 See Proceedings ICM 1932, Vol. 1, pp. 76–77; my translation.

3 See Proceedings ICM 1932, Vol. 1, p. 190, and Proceedings ICM 1932, Vol. 2, p. 19.

**Encyclopedias.** Before focussing on mathematics and mathematicians, let us recall that 'Consolidation and Unifcation' during the thirties was of course not reserved for mathematics, and that it could take on a wide spectrum of political colors. This is conveniently illustrated by encyclopedic projects various sorts of which fourished in those years.4

Among the examples following the classical model of alphabetically ordered learned dictionaries, there were monumental projects destined to express political regimes established after Word War I. The earliest and most impressive case of the genre is the (frst edition of the) *Large Soviet Encyclopedia*; the sixty-six volumes of its frst edition appeared between 1926 and 1933. The mathematician Veniamin Kagan (1869–1953), who held the chair of Geometry at Moscow University as of 1923, was part of the initial founding group of this project.5

Preparations for the Italian *Enciclopedia Italiana di scienze, lettere ed arti*, commonly known as *La Treccani*, also go back to the twenties, under the direction of the industrialist Giovanni Treccani (1883–1961) and Giovanni Gentile. The twenty-fve volumes appeared between 1929 and 1937. The article on 'Fascism' in volume 14 "was written about 1931, signed by Mussolini but obviously composed by a number of hands including that of Gentile."6

An altogether diferent kind of project, and a token of the thirties' search for national orientation outside of Marxist-Leninist or fascist infuence, was the *Encyclopédie française* initiated in 1932 by the French Minister of Education Anatole de Monzie (1876–1947) in collaboration with the historian Lucien Febvre, who had left Strasbourg in 1933 for a chair at Collège de France in Paris. Where the Soviet Encyclopedia and the Treccani negotiated their take on civilization with their respective regimes, Febvre's concept aimed at a modernist7 tableau of the world and of humanity. The reader is not ofered a sequence of quintessential entries in alphabetic order, but a collection of signed systematic expositions of problems and potential answers. Some of these essays are of a fairly technical nature. The whole product was realized as a loose leaf edition, ostentatiously open to improved later versions, so as to avoid the illusion of a fnished, coherent account. The economic success of the undertaking left much to be desired.

The frst volume of the *Encyclopédie française* is devoted to the 'Toolkit of the mind' (*Outillage mental*); its third (and last) part is dedicated to mathematics. A frst edition of it appeared in 1937, edited by Paul Montel.8 The chapters of this frst edition

4 The (German version of) Felix Klein's Encyclopedia of the mathematical sciences—see Section 1.3.3—also continued to produce fascicles in the 1920s and 1930s. However, it was no longer the vector of Mathematics International it had been before World War I. Therefore we will not go into the late phase of the mathematical *Enzyklopädie*. See, however, [Siegmund-Schultze 1993], pp. 98–101.

5 Cf. [Mazliak 2018].

6 See [Smith 1969], p. 412.

7 Say, in the sense of Charles Baudelaire's 1863 essay *La modernité*: "By modernity I mean the transitory, the fugitive, the contingent which make up one half of art, the other being the eternal and the immutable." See [Baudelaire 1964], p. 13.

8 Later additions by new authors, notably Jean Leray (1906–1998), were published in 1950.

were mostly penned by mathematicians of the older generation. One of the oldest among them, Hadamard, wrote in particular an introduction to mathematics which flls the frst 110 columns of this part. After initial remarks of a more philosophical nature, the portrait of our science is loosely structured according to historical periods and their contributions to the shaping of various parts of mathematics. Poincaré's contributions are duly underlined; but Hilbert's formal axiomatics are prominently mentioned as well, and the concluding paragraphs allude to Fréchet's "abstract" spaces as well as to group theory, citing Galois of course, but also Sophus Lie (1842–1899). Hadamard also coordinated the more detailed essays on Analysis. These and the other texts on major subdisciplines of mathematics were mostly written by mathematicians of older generations: Ernest Vessiot (1865–1952), Élie Cartan, Émile Borel, Paul Montel, Maurice Fréchet, Jean Chazy (1882–1955), René Gosse (1883–1943), Arnaud Denjoy, and the Belgian algebraic geometer Lucien Godeaux (1887–1975) were all born between 1865 and 1887. The only younger authors were the Hungarian Béla Kerékjártó (1898–1946), as well as two youngsters: René de Possel (1905–1974) and Claude Chevalley who were also part of Bourbaki in the 1930s.

The various sections have been written in gradually increasing difculty so that the reader can go just as far as his knowledge and energy permit. It is well known that there are no "royal roads" into mathematics. Whoever ventures in this domain will inevitably bump his feet against stones along the way. But we have at least tried to indicate the roads which are the most direct and have the best layout, to discover this whole country.9

All the projects mentioned so far were national enterprises. A totally diferent, inherently international project of an *International Encyclopedia of Unifed Science* (IEUS) was launched by the politically active economist and philosopher Otto Neurath (1882–1945), and further developed in the 1930s at international congresses inspired by the philosophy of the Vienna Circle. Only a torso10 of this project would fnally be published in twenty monographs which were meant to lay the ground for a unifed science based on the principles of logical empiricism.

It would be interesting—although clearly outside the scope of the present book to compare certain texts of this Encyclopedia with parts of Febvre's *Encyclopédie française*, such as for instance Maurice Halbwachs's essay "The numerical point of view" about how statistics provides evidence of social facts, which nonetheless remain extremely hard to interpret.11

Instead, let us just mention in passing that a fair number of mathematicians were part of a committee for the *International Encyclopedia of Unifed Science* instituted at the Paris International Conference for Scientifc Philosophy in September 1935: Élie Cartan, Federigo Enriques, Fréchet, Hadamard, Jan Łukasiewicz (1878– 1956), Richard von Mises, and also Bertrand Russell (1872–1970). No-one from

9 From Montel's preface to the Part on Mathematics of *Encyclopédie française*, p. I•50–7; my translation.

10 Cf. [Dahms 1999].

11 See [Halbwachs & Sauvy 2005]. For a presentation of the issue from the mathematical point of view, see [Brian & Jaisson 2007].

the Bourbaki group seems to have been involved with the IEUS. Claude Chevalley did give a lecture at the 1935 Congress in Paris, but the philosophical view on the language of science he presented there difered substantially from the perspective of the Vienna circle.

#### **6.1 Emmy Noether's Legacy**

Today Emmy Noether (1882–1935) has become a household name, even somewhat beyond the mathematical community, and there are now well-informed books in English that mirror up-to-date research about this exceptional mathematician and woman.12 Readers may know that, in terms of mathematical achievements, she is most famous (a) for her two theorems in theoretical physics of 1918 which link symmetries, i.e., invariance under the action of a Lie group, to physical conservation laws, (b) for her work in abstract algebra, especially ring theory, in the 1920s, and (c) for playing an important part in applying the theory of division algebras to number theory in the late twenties and early 1930s. However, her impact on the development of mathematics between the World Wars was both broader and more profound than any list of theorems can convey.

Since Emmy Noether was a woman, her education and academic career in Germany was ridden with administrative hurdles, and often further complicated by anti-Jewish or political prejudices. Even infuential men like Hilbert could not obtain for her the right to teach at Göttingen University before the end of World War I.13 The position she fnally obtained in Göttingen was but a caricature of her scientifc standing; it was inferior to Sofya Kovalevskaya's Stockholm professorship about 40 years earlier (Section 1.1.7). The "only time Emmy Noether, the mother of modern algebra, was treated by a German authority just as her distinguished male colleagues, and not according to her inferior hierarchical position, occurred when the Nazi government put her on leave, forbidding her to teach at the mathematics institute of Göttingen University, by way of a telegram dated 25 April 1933."14 See Fig. 6.1 for a page from the ministerial fle about her dismissal in 1933.15

12 See the two overlapping books [Rowe & Koreuber 2020] and [Rowe 2021]. These books will point the reader to the vast existing literature on many aspects of Emmy Noether's life and work that we will mention only in passing in our account.

13 Emmy Noether, as a woman, was denied the right to obtain the *Habilitation*. The afair has been analyzed in [Tollmien 1990]; it is retold in English in [Rowe & Koreuber 2020], Section 2.1. Two volumes of Tollmien's new Emmy Noether biography adding new aspects to the story are expected to appear in 2021.

14 Opening sentence of our marginal note [Schappacher & Tollmien 2016], which documents Hermann Weyl's unsuccessful attempt in 1932 to get Emmy Noether elected into the Göttingen Academy of Sciences.

15 Overview of the questionnaire shown in Fig. 6.1. 1) Name, 2) Given name; 3) Date of birth: *23 March 1882*; 4) Nationality (current / at birth): *Bavaria*; 5) Date of habilitation: *4 June 1919*; 6) Career details; 7) Front line fghting in the World War; 8) Race of the four grandparents: *Jewish*; 9) Father or son killed in action in the World War: *No*.

**Fig. 6.1** Emmy Noether's personal questionnaire of 19 April 1933, flled in by herself, with two female gender symbols added by the administration. Credit: [Arch. GStA].

In this section I argue that Emmy Noether's way of practicing mathematics was in fact a model for that 'Mathematical Consolidation and Unifcation' which characterized the 1930s. To see this one has to take a closer look at her personal approach to mathematics. Above we have called her the "mother of modern algebra." This is fair enough as a brief marker, but it could occlude the broadness of her infuence; we need further questions to put us on track.

#### **6.1.1 What is 'Modern' about 'Modern Algebra'?**

A few months ago at a dinner, I asked a colleague from the humanities which label he would see ftting to characterize the 1930s. His immediate answer was: *modernism*. He explained his reaction by pointing to the omnipresence of certain art forms, in particular in architecture, even across very diferent political regimes at the time. Above I have also called Febvre's *Encyclopédie* 'modernist', albeit with a fashback all the way to Baudelaire. There is actually a certain ambiguity among historians of mathematics when it comes to extending the notion of modernism or modernity to the 1930s. For instance, Jeremy Gray's book on the subject [Gray 2008] stops before 1930, whereas for Leo Corry in [Corry 2004] both the Bourbaki enterprise and the slightly later takeof of category theory constitute the very climax of modern algebra with its central concept of *structure*.

Recall from our brief discussion of Mehrtens's book [Mehrtens 1990] at the beginning of Chapter 2 that the most visible radical manifestation of modern mathematics was Hilbert's formal axiomatic approach to the *Foundations of Geometry* in his seminal book [Hilbert 1899]. So this notion of modern mathematics dates back to the turn of the century. However, Hilbert did not care much for the new algebra in his own publications; for instance, "Hilbert's *Zahlbericht* of 1897 made even less use of unifying notions from abstract algebra than one might have expected from a text written in the last decade of the nineteenth century."16 Nonetheless, what Hilbert had done to geometry was heeded, for instance, by the Chicago school of Eliakim Hastings Moore (1862–1932) who transferred Hilbert's paradigm to the comparative study of a great number of diferent algebraic structures in the early years of the twentieth century.17 Outstanding American mathematicians such as Leonard Dickson, Oswald Veblen, and Joseph Wedderburn (1882–1948, originally from Scotland) were in touch with this movement.

This frst decade of the twentieth century is naturally included in *all* historical accounts of modernism, inside as well as beyond mathematics. But it is also sometimes considered a period of "unhappy modernity"18, with mathematicians showing signs of "anxiety"19. This may incidentally remind us of the conficting associations provoked by the thirties.

16 Quoted from [Schappacher 2005], p. 704, where the statement is feshed out further, and Minkowski's diferent attitude is cited.

17 See Section 3.5 on "Postulational Analysis in the USA" in [Corry 2004], pp. 172–182.

18 See the section *Une modernité malheureuse* in [Charle 2011], pp. 335–336. Incidentally, this conclusion of his Chapter 11 follows Charles's brief account of Daniel Halévy's somber, nietzeschean utopia *Historie des quatre ans 1997–2001* published in 1903, in which an epidemic kills 400,000 Parisians in three months, seemingly opening up a better future (for the survivors) based on science and a healthy diet; but this hope is then destroyed in turn.

19 See in particular the analysis in [Gray 2004] of Oskar Perron's 1911 Tübingen lecture "On Truth and Error in Mathematics." Cf. [Gray 2008], Section 4.8.3, pp. 274–277, and the doubts expressed in [Schappacher 2012], pp. 234–235.

Other developments of modern algebra, before Emmy Noether left her mark on the budding discipline, include Ernst Steinitz's (1871–1928) systematic investigation of the (algebraic) feld concept, published in 1910. The interest in studying this abstract notion of algebra was prompted in particular by Kurt Hensel's (1861–1941) introduction of -adic numbers, which had produced a whole bunch of concrete examples of felds never encountered before. Shortly afterwards Adolf (Abraham) Fraenkel (1891–1965) started to investigate ring theory, coining as it were the algebraic notion of ring.20 So modern algebra as a systematic study of algebraic structures existed before Emmy Noether joined in. It was driven by known *instances* of such and such structures—groups, rings, modules, felds—that had previously been researched in more concrete mathematical contexts, typically in the theory of equations, in geometry or in number theory.

In order to fathom Emmy Noether's role for modern mathematics, one has to pin down the kind of insights that the investigation of algebraic structures can yield. Modern Algebra obviously opens up a whole new kingdom of creatures, and thus the modern algebraist would appear like a new Carl Linnaeus, all busy inspecting and classifying even the most exotic genus in the new zoo, all the way from magmas to structures with inner and outer operations. Clearly some sort of stock taking had to be done and in certain cases this represented a formidable mathematical research program in itself—think of the Classifcation of Finite Simple Groups. But if this were all that is meant by modern algebra: a discipline that studies and classifes algebraic structures, calling Emmy Noether the 'mother of modern algebra' would be misleading.

Already Steinitz proceeded diferently in his abstract theory of felds: mindful of traditional problems which could be seen to have involved certain types of felds such as the resolution of polynomial equations—he checked in which way and how far the traditional theory could be generalized, and where it had to be modifed in view of the new supply of structures, for instance for felds of fnite characteristic. This new vantage point for looking at traditional results also allowed Steinitz to criticize Kronecker for having blurred the distinction between splitting a given polynomial into a product of linear factors, as opposed to splitting all polynomials, i.e., constructing the algebraic closure of a given feld.21

Following Steinitz, Emmy Noether's contributions to modern algebra were oriented towards *either separating* aspects that had been previously undistinguished, *or else unifying* seemingly unrelated, or at best analogous theories, by accommodating them under the joint roof provided by a more general abstract theory.

20 Cf. Chapter 4 of [Corry 2004].

21 See [Steinitz 1910], pp. 169–170.

Her relentless ambition to unify shows for instance in her 1919 *Report on the arithmetic theory of algebraic functions of one variable, in its relation to the other theories and to the theory of number felds*.22 She was asked to write this report for the German Mathematical Society in order to fll a deliberate gap which her father and Alexander Brill had left in their colossal report on the theory of algebraic functions written almost thirty years earlier.23 Emmy Noether's exposition is not only more concise than the commendable scholarly prose of Brill and her father; she also pushes the juxtaposition of parallel theories to new levels of formal compatibility. All she was supposed to deliver was an account of various theories developed by others decades ago. However, while being even more careful than usual to always refer to the original papers, Emmy Noether also arranged things in a new, unifying manner.

It was well known at least since the 1880s that the arithmetic theory of number felds—i.e., fnite extensions of the feld Q of rational numbers—is in many ways analogous to that of function felds of one variable—i.e., fnite extensions of the feld C().24 The *relation*25 between the two cases is one of the foci of Emmy Noether's 1919 report. Instead of moving back and forth between the two cases, she tried to bind them together cogently under the umbrella of a single more general theory. However, this unifying theory did not really exist yet; its fully fedged exposition would be the object of her 1927 paper *Abstract Setup of the Theory of Ideals in Number Fields and Function Fields*.26

This later paper incidentally reminds us of the strong international dimension of modern algebra. Emmy Noether was not the only mathematician trying to develop a general theory of ideals in commutative rings; some of her insights had actually been anticipated by the Japanese mathematician Masazo Sono (1886–1969). She

22 See [Noether 1983], pp. 271–292. – The reproduction of this report is an instance of sloppiness in the edition of Emmy Noether's *Collected Papers*: The title printed on p. 271 misses the last 8 words, and on the fnal p. 292 the last footnote of her report is not reproduced. Other mistakes which I stumbled across occur on p. 560, where the text reproduced is not that of her Bologna ICM talk, and on p. 636, where the printed title gets a word wrong.

23 See [Brill & Noether 1892–93]; Kronecker's sudden death is cited to explain the omission of the arithmetic theories from this report on pp. I–II.

24 Both a proposal for a unifed treatment of the two cases by Leopold Kronecker and a treatment of the second case modelled on the frst one by Richard Dedekind and Heinrich Weber were published in *Crelle's Journal* in 1882; for more background and details see for instance the discussion in [Schappacher 2010], pp. 3262–3267.

25 The German word *Beziehung* is in the title Emmy Noether chose for her report.

26 See [Noether 1983], pp. 493–528. The paper was written in 1925 and appeared in *Mathematische Annalen* in 1927.

eventually realized that she had had this precursor, and in her 1927 paper she refers at four places in detail to various articles by Sono published in English in the *Memoirs of the College of Science, Kyoto Imperial University* between 1917 and 1924.27

From the point of view of the genealogical thinking popular in parts of the mathematical community, Sono was the academic grandfather of two Fields medalists. Sono's student Yasuo Akizuki (1902–1984) counted among his own numerous thesis students Heisuke Hironaka (b. 1931; Fields Medal 1970) and Shigefumi Mori (b. 1951; Fields Medal 1990).

Anyway, reading her 1919 report we see Emmy Noether at work at a time when she could not simply allude to general notions of what we call today *noetherian rings* or *Dedekind rings*. Nonetheless, she was resolutely looking at things from the superior vantage point of such a unifying theory. Let us give an

**Example.** Consider the elementary statement about polynomials which says that if () = ( − ) · () with () ≠ 0, then the derivative ′ () is divisible by ( − ) −1 [and by no higher power of ( − ), unless we are working over a feld of characteristic with dividing ]. Somewhat analogous statements were known from the so-called theory of ramifcation, involving the 'discriminant' and 'diferent' of an algebraic number feld, or a function feld.

In Section 2 of her report, entitled *Parallelism and Diferences in the Theory of Ideals of Algebraic Numbers and Functions*, Emmy Noether does not simply review, but tries to *explain* apparent diferences of ramifcation theory between the two cases: "*Diferences* in the further development [of the theory] are occasioned by special properties of the ground feld. Thus the fact that in the case of algebraic numbers *elements of the ground feld are at the same time exponents* implies that the ramifcation ideal may be divisible by a higher power than the ( −1) th of a prime ideal ℘ whose -th power divides precisely (in the case where is divisible by )."27

In other words, she explains the absence of the so-called wild ramifcation in function felds (of characteristic 0) by the fact that positive numbers are invertible in C ∗ . But again, the general theory afording this explanation was not fully worked out yet at the time of the report. She would present it in a talk at the Prague meeting of the German Mathematical Society in 1929.28 There she repeated that her approach goes "beyond formal analogies." Indeed, she developed an intrinsic reformulation, purely in terms of ideals, of the elementary statement about polynomials quoted above. And she gained signifcantly

27 It would seem that she was not yet aware of Sono's work when she published her momentous frst paper on the theory of ideals in *Mathematische Annalen* 83 (1921), see [Noether 1983], pp. 354– 396. Cf. [Kümmerle 2021] who mentions Sono's work against the background of the establishment of Japanese mathematics, including mathematical journals.

#### 6.1 Emmy Noether's Legacy 163

greater generality; the "most essential result" of her work lay in those "completely invariant structure theorems which are also valid for algebraic functions in several variables."29

This example illustrates how "modern algebra" in the hands of Emmy Noether instead of just introducing a plethora of novel structures—invariably strove towards a parsimonious hierarchy of mathematical theories. It is this peculiar explanatory power of generality which was acknowledged by Albert Einstein when he wrote to Hilbert in 1918 about Emmy Noether's work on diferential invariants: "It impresses me that one can view these things from such a general standpoint."31 Writing anachronistically one could say that Emmy Noether lay the ground for Alexander Grothendieck's unifcation of commutative algebra and algebraic geometry.

The basic toolkit for the modern, unifying expeditions into the thicket of traditional mathematics would quickly be covered in textbooks and monographs of which the outstanding example was Bartel L. Van der Waerden's *Modern Algebra, based on lecture courses by E.*[mil] *Artin* [1898–1962] *and E.*[mmy] *Noether* in two volumes.32 The frst edition appeared in 1930–1931. The special attention to reorganizing theories with respect to one another in this modern way of doing mathematics is crystallized at the beginning of the book in the *Leitfaden* that shows the various chapters and outlines their logical dependence in a fowchart.

At the beginning of the introduction, Van der Waerden ofers without commitment various adjectives in quotation marks to characterize the new way of doing algebra: "abstract", "formal", "axiomatic." In the troubled times ahead these words could easily acquire a pejorative connotation. Hermann Weyl had to grapple with this in the petition letter which he contributed to the vain efort of stopping Emmy Noether's dismissal in 1933:

She represents principally "abstract algebra." In this context the word 'abstract' does not at all indicate that this branch of mathematics is especially remote from life (*lebensfern*). The predominant tendency is rather to master the problems by visionary thought, by arranging a conceptual framework as appropriate as possible to the subject matter, instead of blind calculation.33

27 See [Noether 1983], p. 278; my translation.

28 A corresponding manuscript was published only after her death and is the last article reproduced in her *Collected Papers*; see [Noether 1983], pp. 690–710.

29 See [Noether 1983], p. 692; my translation. The word 'invariant' here means that these theorems of modern algebra avoid any case distinctions according to the intended setting, like for algebraic numbers or algebraic functions. Cf. Nathan Jacobson's comments on this paper in [Noether 1983], pp. 15–16.

31 Quoted from [Rowe 2021], p. 74.

32 See [Van der Waerden 1930–31].

33 From Emmy Noether's ministerial fle kept at *Geheimes Staatsarchiv Berlin-Dahlem*, p. 39. Weyl's letter is dated 12 July 1933. Cf. [Rowe 2021], p. 145, who points to Zermelo's plea in 1929 to replace the expression "abstract algebra" by "general algebra."

**Fig. 6.2** *Leitfaden* from Van der Waerden's book *Moderne Algebra*, frst edition.

As of the fourth edition of Van der Waerden's extremely successful textbook, in 1955, the epithet "modern" was dropped from the title. Van der Waerden motivates this in his 1955 preface by quoting from a book review by the algebraist Heinrich Brandt (1886–1954) who rejected the term 'modern' as being associated with fashions that come and go. Similarly, ten years later, Serge Lang (1927–2005) opened the preface to his textbook on *Algebra* by the following quote of Severi from 1949—certainly one of the more unlikely choices of a mathematician to quote about modern algebra:

I prefer to call it 'abstract algebra' rather than 'modern algebra' because it will undoubtedly live for a long time and will thus end up becoming the ancient algebra.34

Both instances refect the fact that after World War II the notion of modernity had largely lost the luster that it still had in the 1930s.35

34 See [Lang 1965], p. V; my translation.

35 See also [Corry 2004], p. 61, who cites the analogous case of Dickson's new edition of his book entitled *(Modern) Algebraic Theories.*

#### **6.1.2 Emmy Noether's** *Aufassung* **and its Infuence**

We have seen that Emmy Noether's approach was about reorganizing mathematics, about new ways to look at theories, theorems, and problems. This is the intrinsic reason for Emmy Noether's broad infuence on mathematical practice between the World Wars, beyond her own theorems.

It seems that Emmy Noether herself was not particularly gifted in explicating her philosophy of mathematical practice.36 There is a word that she often used in this context: *Aufassung*. It is a rather poor choice for a word expressing a methodological tenet, because it has many plain meanings: opinion, point of view, conception, etc. This makes it quite a challenge for the translator, for example when she speaks about her *Arbeits- und Aufassungmethoden*:

My methods are methods about how to work and how to view things; this is why they have penetrated everywhere, anonymously.37

We have already mentioned in Section <IP_ADDRESS> that the word *Aufassung* made it into the title of her ICM talk at Bologna in 1928. And in a letter to Hasse about her preparation for the 1932 plenary ICM talk in Zürich, she tells him that, for once, she has read Gauss, adding the peculiar remark that she "learned a lot from Gauss about *Aufassung*":

I learned a lot from Gauss about how to view things; above all, that it is good to place the verifcation of the fact that the classes determined by factor systems are ray classes, at the end . . . 38

Never mind the mathematical details; the quote shows that *Aufassung* expresses itself in proof arrangements. In her Zürich lecture she explained how she managed to smuggle non-commutative algebras into a study of (commutative) felds:

*With the help of the theory of algebras one looks for invariant, simple formulations of wellknown facts about quadratic forms or cyclic feld extensions, i.e., formulations that only depend on structural properties of algebras. Once such formulations have been proved . . . a transfer of these facts to general galois feld extensions is achieved.* . . . the passage to the non-commutative world is aforded by *viewing* feld and group *simultaneously*, via their "crossed product."39

We learn from this quote that the rearrangements of proofs in the interest of the most adequate, 'invariant' insight, or *Aufassung*, can require the introduction of abstract intermediary objects used to assemble and treat data provided by the

36 See for instance Paul Dubreil's description of an unsuccessful presentation of hers before a mixed audience of philosophers and mathematicians in Hamburg in 1930; Dubreil's account is translated in [Rowe 2021], p. 152.

37 From a letter to Hasse of 12 November 1931; see [Lemmermeyer & Roquette 2006], p. 131; cf. their comments in endnote 8, pp. 132–133, and their proposed translation p. 8. The translation given here is mine.

38 We quote from the translation in [Goldstein, Schappacher, Schwermer 2007], p. 557, where the original text is also given. The quote marks the end of Lemmermeyer's contribution to the book.

39 Proceedings ICM 1932, vol I, p. 189; the italics are Emmy Noether's; my translation.

original problem. This is Emmy Noether's philosophy of mathematical practice in a nutshell. It is also the blueprint for seminal mathematical innovations during the interwar period. For instance, Emmy Noether encouraged Van der Waerden to start rewriting Algebraic Geometry from the point of view of her algebraic approach. This would become the frst among several diferent fundamental rewritings of Algebraic Geometry in the twentieth century.40

In 1935, shortly after Emmy Noether's death at Bryn Mawr, Pennsylvania, and around the time of the frst Bourbaki congress, Claude Chevalley wrote an essay about *Variations of mathematical style*. Even though the collective project is not explicitly mentioned, this text is in fact the frst manifesto of Bourbaki, written and published even before the group had adopted their collective pen name.41 It faithfully follows Emmy Noether's legacy, even though it is Hilbert's name, not hers, that Chevalley uses to label the new *style* of mathematics.

**Fig. 6.3** Group picture of the Bourbaki Congress at Besse la Chandesse in July 1935. Standing from left to right: Henri Cartan, René de Possel, Jean Dieudonné, André Weil; Luc Olivier (biologist). Sitting: A. Mirlès, Claude Chevalley, Szolem Mandelbrojt.

Two recent mathematical practices (*styles*) are confronted in Chevalley's essay: Weierstrass's "style of the epsilons" from the second half of the nineteenth century which achieves impeccable rigor at the price of reducing all statements to relations involving real numbers—is opposed to the freedom of the "Hilbertian mathematician"42 whose objective it is to prove each theorem in an intrinsic way, i.e., without

40 See [Schappacher 2007].

41 See [Chevalley 1935]; the essay appeared in the Fall of 1935.

42 We borrow this expression: *mathématicien hilbertien*, from [Chevalley & Dandieu 1932], p. 111. This earlier paper justifes the existence of an active mathematician with respect to the *personalistic* philosophy of Arnaud Dandieu (1897–1933) and Alexandre Marc (1904–2000) and its supreme value of creativity. That Chevalley was strongly committed to this thought in the 1930s shows in the more than 30 philosophical and political essays he published between 1932 and 1938.

introducing auxiliary objects or making arbitrary choices (like choosing a coordinate system to translate a geometric problem into a statement about real numbers) that are not germane to the statement at stake:

. . . constructive defnitions from real numbers, even if they made rigorous proofs possible in the frst place, often had the efect of seriously hiding the very nature of what they strove to defne, or to unduly blur mathematical domains that are actually distinct. This leads to futile complications that can be observed in many classical proofs. They arise when methods are used that have nothing in common with the desired result, or one could say: methods which do not have the same transformation group as the result.43

The agenda of the Hilbertian mathematician as described by Chevalley involves precisely the sort of reorganization of mathematics which was Emmy Noether's central concern.

Today almost all mathematical facts are categorized in more or less autonomous theories that have accumulated. The principal examples are the following: in algebra one fnds the general theory of felds . . . ; group theory; the theory of those systems having less restrictive algebraic relations than those of felds (like rings or algebras)—these are essentially due to Emmy Noether whose recent passing is deplored by the mathematicians—; in analysis, the theory of measure and integration, of topology, of general Riemann surfaces, Hilbert spaces, etc.; in geometry, projective and conformal geometry; the theory of Riemann spaces; combinatorial topology . . . One should not imagine that these theories exist in isolation from each other. On the contrary, they are coalescing into more complex theories. For example, if one operates on certain elements between which both strictly algebraic relations (like multiplication) and relations of continuity exist, one obtains topological algebra, which provides richer properties than the juxtaposition of algebra and topology. There are numerous such complex coalescences of theories. On the other hand, very diferent theories may be transposed into one another provided they are based on the same axiomatic structure: for instance, probability calculus and measure theory.44

This new ordering of mathematics—and the energetic way in which it was practiced by Emmy Noether in her research, and presented here by Chevalley—is a far cry from any idea of crisis. It is not the kind of radical overhaul of mathematics that the foundational crisis of the 1910s and 1920s had suggested to mathematicians like Brouwer and Weyl—see Section 2.2. Ongoing research work on the one hand and an increasing number of systematic presentations and textbooks on the other including the enormous Bourbaki project—see Section 6.3 below—would completely remodel the world of mathematics from the inside.

#### **6.1.3 Emmy Noether's International Network**

The nineteenth century had professionalized science, and mathematics. In Part I of this book we have traced international aspects of this evolution. For instance, in the diferent editions or translations of Felix Klein's Encyclopedia, we were looking for interesting comparisons of national mindsets. The 1930s were diferent. Emmy

43 See [Chevalley 1935], p. 379–380; my translation.

44 See [Chevalley 1935], p. 383; my translation.

Noether's *Aufassung* prepared the way for fundamental mathematical renovation in the context of international teams of mathematicians, be that in Algebra, Topology, Arithmetic or Geometry. Professionalization continued, but diferently, with stipends, jobs and institutes created by national or private sponsors. The Rockefeller Foundation, which we have presented in Section 5.1, was but one of several donors. The growing mobility options of young researchers started to show its efects on the world of mathematics. The founding fathers of Bourbaki for example profted from various stipends or invitations acquainting them with global mathematical centers of the time. These modern instruments of Science International would merge with the novel structural emphasis inside mathematics to build a new way of living mathematics which involved more direct personal interaction than ever before. The resulting consolidation and unifcation was the work of many, and it was an international enterprise.

. . . the research of Fräulein Noether and her students, van der Waerden, [Heinrich] Grell [1903–1974], [Wolfgang] Krull [1899–1971], who disseminate the theory of ideals all over algebra; research on the theory of sets and dimension ([Felix] Hausdorf [1868–1942], [Karl] Menger [1902–1985], Urysohn and the whole Polish school), on group theory ([Otto] Schreier [1901–1929], [Élie] Cartan). In all these domains only the use of axiomatics allows a perfect adjustment of the proof to its object. . . . 45

Chevalley's list of (groups of) mathematicians is euro-centric but extends beyond algebra. Emmy Noether fts into this picture, but her mathematical networking between 1925 and 1935 extended further, both geographically and in terms of mathematical domains. The group around Emmy Noether has been called an "International School in Modern Algebra."46 She did build a school of modern algebra, and it was international. But just as the title "the mother of modern algebra" can mask her broader infuence on mathematics, also her international contacts reached beyond the group of her (former) students.

Pavel Alexandrov, one of the designers of contemporary topology whom we have briefy mentioned in Section 5.1 above, was a frequent visitor to Göttingen in the 1920s, accompanied by his partner Urysohn until the latter's tragic death. A close friendship developed between Emmy Noether and Alexandrov. The latter arranged for Emmy Noether to stay in Moscow during the winter of 1928, following the Bologna ICM. Her lectures on Algebra there, given in German, provided direct inspiration for Alexandrov's own teaching and left their mark on the Moscow mathematical community.47

Alexandrov also formed a troika of friendship with Heinz Hopf (1894–1971) and Otto Neugebauer (1899–1990). As topology was in the air, the three of them adopted a simplex as a symbol of their friendship. This pops up time and again in their correspondence.

45 See [Chevalley & Dandieu 1932], p. 105; my translation.

46 For instance in the title of the ffth chapter of [Rowe 2021].

47 See [Rowe 2021], pp. 131–133.

Another important connection of Emmy Noether's would reach out to Japan. These contacts were mediated by Teiji Takagi whose work on Hilbert's 12th problem and class feld theory became well-known in Europe in the 1920s.48 Takagi was one of the vice presidents of the Zürich ICM in 1932, and a member of the committee for selecting the frst Fields Medals to be awarded in 1936.

Two of Takagi's students, Zyoiti Suetuna (1888–1970) and Kenjiro Shoda (1902– 1977), spent time in Göttingen in the 1920s. Suetuna—who would succeed his teacher Takagi on his chair at Tokyo University in 1936—was more interested in analytic number theory. Shoda on the other hand would become one of Emmy Noether's students, adapting as well as possible to her way of teaching, talking, and doing mathematics.

They remained in contact from a distance after 1929, when Shoda returned to Japan. Soon afterward, he began to write his textbook *Abstract Algebra*, which was frst published in 1932 and then reprinted many times later. Its impact on mathematics in Japan has sometimes been compared with that of van der Waerden's *Moderne Algebra* on research in Europe and the United States. Kenjiro Shoda went on to a highly successful career, beginning in 1933 with his appointment as professor in the Faculty of Science at the newly founded Osaka University.49

In the Summer of 1933, Takagi and Shoda wrote letters of support for Emmy Noether to the ministry as part of the concerted efort to avoid Emmy Noether's dismissal. Takagi in his letter of 12 July 1933 insists on the great loss that her dismissal would mean to German science, and he adds, obviously alluding to the racist politics threatening Emmy Noether, that the loss would be all the more deplorable "as it would be due to a circumstance for which she is not to blame."50

Apart from the 14 petitions written by distinguished colleagues on behalf of Emmy Noether, which were sent that summer to the German ministry via the administration of Göttingen University, there was also a letter of her students which refects the Göttingen political constellation at the time in tragic irony: The letter chooses to explicitly endorse the "National Revolution" of the Nazis in order to make the authorities well-disposed to the request. The students thus insist that "it is not by chance that all her students are Aryan; this is grounded in her *Aufassung* of the essence of mathematics which is especially germane to the Aryan way of thinking."51 As it happened, among those who signed the letter were two young mathematicians from China: Chiungtze C. TSEN (1898–1940) and Wei-Liang CHOW (1911–1995).52 Under Emmy Noether's guidance, Tsen had just proved that a function feld in one variable over

48 Cf. [Schappacher 1998].

49 See [Rowe 2021], p. 128, and more generally the vivid, partly anecdotal description of Shoda's apprenticeship with Emmy Noether on pp. 125–128.

50 From Emmy Noether's ministerial fle kept at *Geheimes Staatsarchiv Berlin-Dahlem*, p. 35: *. . . zumal wegen eines Umstandes, woran sie keine Schuld trägt*.

51 From Emmy Noether's ministerial fle kept at *Geheimes Staatsarchiv Berlin-Dahlem*, pp. 13–14. Cf. [Rowe 2021], pp. 200–201. The letter is undated but was surely submitted in June 1933.

52 I tend to follow the version of a Chinese name which is most commonly used in English publications; on the frst appearance, I HIGHLIGHT the family name.

an algebraically closed feld admits no non-commutative fnite-dimensional central division algebras. Formally he got his doctorate only after Emmy Noether's emigration to Bryn Mawr.53 Chow had only recently come to Göttingen and found the Mathematical Institute he was keen to study at in political agony—see Section 6.4 below.

#### **6.2 Encounters, Workshops and Congresses in the 1930s**

In the previous section we have focussed on Emmy Noether in order to present key elements of the mathematical renewal that asserted itself during the 1930s. The new era not only changed what mathematicians would know, or would have to learn, but how they worked, and indeed how they spent their days. Get-togethers outside of one's professional institution, for the immediate exchange of mathematical ideas and results, became more and more frequent; both individual face-to-face meetings and bigger conferences. We have already stressed at the beginning of Section 6.1.3 the increased mobility of young researchers which facilitated international exchange on an unprecedented scale.

A meeting with the Polish mathematician Juliusz Schauder [1899–1943] reveals to [Jean] Leray he topological techniques required by his thesis. The consequence is a joint paper also published in 1934, worked out in two weeks, the year before, in Luxembourg's garden of Paris. Topological degree theory in infnite-dimensional Banach spaces is born, as well as the global theory of nonlinear elliptic partial diferential equations. The Leray-Schauder degree and the Leray–Schauder continuation method remain the model for the whole development of nonlinear functional analysis and of fxed point theory. The founding paper, whose style is astonishingly modern, is one of the most quoted and used mathematical works in the twentieth century.54

Schauder had been able to come to Paris on a Rockefeller stipend.55 His frst meeting with Leray was mediated by Hans Lewy (1904–1988)—another mathematician who had fed Göttingen for political reasons—and took place over lunch in a modest restaurant in rue Soufot, close to the *Panthéon* and *Institut Henri Poincaré*. As for Leray, he received a grant from the French *Caisse Nationale de la Recherche Scientifque*, the predecessor of today's CNRS (where only the initial letter 'C' has changed its meaning; it now stands for *Centre*). This allowed him to attended courses in Germany, at Berlin and Leipzig. During their joint work in Paris and in their subsequent correspondence Schauder and Leray would mostly converse in German. In 1935 Leray was invited to give the distinguished *Cours Peccot* lectures at *Collège de France* on the topological degree.56

53 See [Ding et al. 1999], which also reviews Tsen's later work in China, before his untimely death. 54 See [Mahwin 1998], p. 200.

55 Cf. [Siegmund-Schultze 2001], *passim.*

56 Cf. [Kantor 2000], as well as the booklet compiled for the event *Jean Leray et les équations aux dérivées partielles* at Université de Nantes 9 November 2006: [URL 09].

#### **6.2.1 Specialized Conferences**

As for specialized research meetings in mathematics—of more than just two coauthors, and outside of regular seminars—probably the frst example that should be mentioned in the 1930s is the workshop on the arithmetic of skew felds held at Marburg in February 1931. It was Emmy Noether who prodded Hasse into organizing it; he had recently succeeded his former thesis advisor Kurt Hensel in the small university town. She jokingly called the gathering a *Schiefkongress*, i.e., 'skew congress.' Jacques Herbrand (1908–1931) from France was one of the young participants.57

That the 1930s were indeed the frst decade of intense international conferences on specifc mathematical subjects is nicely documented by the following passage from a brief overview of mathematical congresses.

In 1933 at Geneva, a series of international conferences and congresses dedicated to the study of specialized research domains was created under the name *Conférences internationales de sciences mathématiques*. In October 1933 a conference on Quantum Theory was held. The subject in June 1934 was Mathematical Logic; the Theory of Electrons in Metals followed in October 1934; Topology in October 1935; Partial Diferential Equations in June 1935; Theoretical Probabilities in October 1937; Applied Probabilities in July1939. This last conference was organized with the help of the International Committee on Intellectual Cooperation58 which in 1938 had brought the "Conversations on the Foundations of Mathematics" to Zürich in 1938. In May 1934 a meeting on Tensorial Diferential Geometry gathered in Moscow. This conference decided to form groups of scholars for a planned research work. The members of each group would stay in touch via scientifc correspondence exchanging interesting problems. In August 1935 at Brussels, the frst International Congress on Mathematical Recreation was held. In September 1935 at Moscow, the First International Congress on Topology was held, at the same time as the International Congress on the Philosophy of Science in Paris. Clearly, the need to bring together specialists from all over the world around specialized problems was strongly felt. Let us hope that this will not bring about a fragmentation of mathematics, but that, on the contrary, following Hilbert's view, the study of a particular subject can but strengthen to whole.59

Following Hollings and Siegmund-Schultze, we may slightly enrich the list, and put things into perspective:

As well as specialised conferences, the 1930s saw several international conferences on a level lower than the ICM, for instance the 8th and 9th Scandinavian congresses (1934 in Stockholm, 1938 in Helsinki), and several conferences organised by the Germans and Italians, even after the outbreak of the war but necessarily with the participation only of their wartime allies. One specialised international conference on applied mathematics not mentioned by Wavre above is the Volta Congress in Italy (1935) on high speeds in aviation.60

The outstanding event among the meetings listed above by the Swiss mathematician Rolin Wavre (1896–1949) was without doubt the 1935 Moscow Congress on Topology. The reasons for this are its truly international format and the timely

57 Cf. [Rowe 2021], pp. 157–159.

58 Or ICIC for short, cf. Section 4.2 above.

59 See [Wavre 1948], pp. 301–302; my translation.

60 See [Hollings & Siegmund-Schultze 2020], p. 145–146, and the further references given there.

presentation of seminal ideas, which would redirect research on Topology and related domains for years to come. It was called the "First International Congress on Topology", but it was also the last such conference that took place in Stalin's Soviet Union.

Topology entered a new era in the thirties, emerging as a mathematical discipline of its own. As Moritz Epple has pointed out, this transpires when one compares the research orientations of topologists like Max Dehn, Kurt Reidemeister, and James W. Alexander. Referring to the well-known lecture about the professional aspects of science (*Wissenschaft als Beruf*) published in 1919 by the sociologist Max Weber, Epple links Alexander's professional situation in Princeton (full professor at Princeton University since 1928, then at the newly founded IAS) to the peculiar modern, impartial overview of various approaches to topology that Alexander surveyed in his plenary lecture at the Zürich ICM in 1932.61 In this well-prepared terrain, the Moscow Congress planted new seeds. We quote from Hassler Whitney's (1907–1989) account of the event:

The International Conference in Topology in Moscow, September 4–10, 1935, was notable in several ways. To start, it was the frst truly international conference in a specialized part of mathematics, on a broad scale. Next, there were three major breakthroughs toward future methods in topology of great import for the future of the subject. And, more striking yet, in each of these the frst presenter turned out not to be alone: At least one other had been working up the same material.

At that time, volume I of P. Alexandrof & H. Hopf, *Topologie*, was about to appear. . . . Its introduction gives a broad view of algebraic topology as then known; and the book itself, a careful treatment of its ramifcations in its 636 pages. (It was my bible for some time.) Yet the conference was so explosive in character that the authors soon realized that their volume was already badly out of date; and with the impossibility of doing a very great revision, the last two volumes were abandoned.

. . .

What was the main import of the conference? As I see it, it was threefold:

1. It marked the true birth of cohomology theory, along with the products among cocycles and cycles.

2. The pair of seemingly diverse felds, homology and homotopy, took root and fourished together from then on.

3. An item of application, vector felds on manifolds, was replaced by an expansive theory, of vector bundles.

Yet seven years later, a single paper of Hopf would cause a renewed bursting open of the subject in a still more general fashion.62

Concerning Whitney's frst item, the omnipresence of cohomological techniques in all branches of geometry, topology, and algebra today is an important new largescale development in pure mathematics of the last 75 years. One might even be tempted to call it a revolution of mathematical practice. It appears that the history

61 See §104, *Topologie als Beruf* in [Epple 1999], pp. 348–352. I would translate the title of this section as "Topology as a profession", although English translations of Max Weber's lecture tend to have "Science as a vocation."

62 From Whitney's contribution to [Duren et al. 1988], which is entitled *Moscow 1935: Topology Moving Toward America*, pp. 97 and 109–110.

of this profound transformation has not been properly studied yet.63 The Moscow Congress was but an early milestone of this development. The cohomological practices have at least two diferent historic roots: the calculus of homologies inherited from Poincaré and the early combinatorial topology on the one hand, and the continuation of Emmy Noether's use of crossed products which would usher in group cohomology on the other.64 Even Alan Turing (1912–1954) briefy tapped into this mathematical oil well in the thirties.65

Given the remarkable importance of the Moscow Congress, people have been interested in reconstituting the lists of participants and lectures presented. Part of the problem is to correctly identify each of the 24 mathematicians—22 men and two women—on the famous group picture.66

**Fig. 6.4** Group picture of the First International Congress on Topology, Moscow 4–10 September 1935. Credit: [Arch. ETH].

63 See, however, the upcoming volume on the history of the notion of *Duality* edited by Ralf Krömer.

64 See for instance the remarks following equation (15) in [Teichmüller 1982], pp. 550–551, on the one hand, and the Appendix A to [Lefschetz 1942], written by Samuel Eilenberg and Saunders Mac Lane, on the other.

65 See [Turing 1938]. I thank Johannes Huebschmann, Lille, for drawing this paper and its relation to Oswald Teichmüller's (1913–1943) cocycle to my attention.

66 For this problem see [Apushkinskaya et al. 2019], pp. 7–9.

Another issue is to name all those that were present at the conference but do not happen to be in the picture.67 For instance, both André Weil and Oscar Zariski were in Moscow during the congress and gave lectures at the University after the meeting, on Algebraic Geometry rewritten in a more modern algebraic language.68 Interestingly, Weil was convinced that he owed his being invited to the congress to the fact that he had spent time with the Noether circle and Alexandrov a few years earlier.69

Having extolled the merits of this landmark Congress, it is only fair to report that not all participants were equally impressed, and some may have missed the important reorientation of the feld that began to transpire in Moscow. Juliusz Schauder for instance reported to Jean Leray—frankly admitting that he himself was not a topologist—to have come away with

the impression that there is currently a standstill in topology and that people are unable to attack fundamentally new problems. Almost all the talks revolved around new ways to defne manifolds, properties of cycles, and so forth, which only roll out broadly (or in the best case, complete) in an uninteresting way what is already known. In general, these investigations are not tailored towards applications and the mathematicians that were present there are interested only in topology as such. Two talks, however, were kind of sensational in that they presented solutions to problems which may have some importance . . . 70

To summarize, a new type of mathematical networking took place in the 1930s, characterized by specialized meetings. Such conferences continue to mark the life of research mathematicians today. Before closing this section, let us review international mathematical congresses of the same decade that were not devoted to a specifc mathematical topic or discipline.

#### **6.2.2 ICMs of the Thirties**

Only two ICMs could be realized after the Bologna ICM of 1928 and before 1950; they took place in Zürich in 1932, and in Oslo in 1936.

In addition to these there was also a curious International Congress organized by the French Mathematical Society (SMF) in connection with the Paris World Fair in July 1937: the *Réunion Internationale des Mathématiciens*. This faint reminder of

67 See [Apushkinskaya et al. 2019], p. 5.

68 See [Parikh 1991], p. 79, and [Audin 2011], p. 473. Cf. Weil's personal account of his stay in Moscow in [Weil 1992], p. 106–110.

69 See [Weil 1980], p. 531.

70 From Schauder's letter to Leray dated 2 December 1935 in [Arch. MIR]; my translation from German. I thank Christophe Eckes, Nancy, for having shared this source with me. The two talks that Schauder singles out—because others had told him of their importance—were given by Georg Nöbeling (1907–2008) and Witold Hurewicz.

the 1900 ICM, which had also taken place during a Paris World Fair, seems to have been decided rather on short notice, in January 1937, by the Council of the SMF. However, the mathematical presence in the 1937 World Fair was not an accident.

In 1933, the French *Commission for Intellectual Cooperation*71 and the *Confederation of Intellectual Workers* (CTI) . . . decided to entrust the General Commissariat of the international exhibition planned in Paris in 1937 to the deputy Adrien Berthod (1878–1944) who belonged to the Radical group. Berthod was a close associate of Borel, himself a deputy since 1924 and vice-president of CTI. Another vice-president of the CTI was . . . André Léveillé (1880-1962). . . . Berthod asked Léveillé to take care of the coordination of the intellectual aspects of the international exhibition. This included a section of Sciences, and in particular a sub-section called 'Scientifc discoveries in their applications' chaired by the physicist Jean Perrin (1870–1942) with Borel as vice-president. A project was proposed, notably under the infuence of the exhibition 'A century of progress' presented in Chicago in 1933, which focused on the presentation of technological advances and on the participation of visitors to discover and understand scientifc matters by touching objects and participating in realtime experiments. . . . The [Paris] project was . . . [a] huge success. . . . [It led to the creation of] the *Palais de la Découverte* ('Palace for discovery'). It should be noted that the initiative was facilitated by the fact that the government in power in 1937 was still the *Front Populaire* government, in which Jean Perrin was the undersecretary of state for research.72

The conference organized by the SMF was also generously fnanced by the French Minister of Foreign Afairs Yvon Delbos (1885–1956). According to the Proceedings of the event, "more than 200" members of the SMF, French or foreign, "adhered" to this meeting.73 Twelve lectures were given and later published in [SMF 1939]. They were due to the Frenchmen Denjoy and André Marchaud (1887–1973)74, the Romanian mathematician Petre Sergescu (1893–1954)—one of the initiators, in 1929, of the Congresses of Romanian mathematicians; in 1937 he spoke on the mathematics in medieval Paris—, the Swiss mathematician and philosopher Ferdinand Gonseth (1890–1975), the Belgian algebraic geometer Lucien Godeaux, the Dutchman Johannes Gualtherus van der Corput (1890–1975), the Italians Tullio Levi-Civita and Vito Volterra (on the Lotka–Volterra equations in biomathematics); Marcel Riesz (1886—1969) of Hungarian origin, Stanisław Zaremba (1863–1942) from Poland, Richard von Mises—at the time based in Istanbul, Turkey, following his emigration from Nazi Germany—, and the British analyst Laurence Chisholm Young (1905– 2000), son of Grace Chisholm Young and her husband William Henry Young.75 In spite of well-known names in this list of speakers, the Proceedings of this meeting do not radiate the same spirit of the thirties we have encountered before.