The Nazis were noted for their sensitivity towards student opinion. This was dramatised in November 1933 by the encounter in Gottingen between the 56-year-old internationally famous Jewish number theorist and professor Edmund Landau and the 20-year-old brilliant student and Nazi Oswald Teichmuller. Teichmuller explained that because Landau was Jewish, he could not appropriately teach Aryan students in their second semester. This was not to be regarded as anti-Semitic, but rather that each student needed to absorb the international-mathematical skeleton from his teacher and to clothe it with his own flesh in accordance with his mental viewpoint. Thus it was acceptable for Landau to teach advanced students because they would have already formed their mathematical mental outlook.

Not surprisingly, the other Nazi students were puzzled by this curious blend of bigotry and differentiation. The episode reflected the generally fraught relationship between National Socialism and mathematics. After all, Hitler had written in * Mein Kampf * : "The folkish state must not adjust its entire educational work primarily to the inoculation of mere knowledge, but to the breeding of absolutely healthy bodies. The training of mental abilities is only secondary... a man of little scientific education but physically healthy, with a good firm character, imbued with the joy of determination and willpower, is more valuable for the national community than a clever weakling." This meant that mathematicians, like scholars in other branches of science, were under pressure to justify their existence in a climate where mental agility was not particularly appreciated.

Mathematicians have been largely ignored in the scholarly and popular literature on the Third Reich - the topic merited only five pages in John Cornwell's book * Hitler's Scientists * . This is why Sanford Segal's * Mathematicians under the Nazis * is so important. It is the first monograph devoted to the social history of mathematics and mathematicians in Germany during the Nazi regime. As the author notes, there is no attempt at a full discussion of the expulsion of Jews and others from academic posts, or of the Holocaust. The emphasis is on the people who remained after the exodus of Jewish talent and also outstanding non-Jews. Some of the Jews who stayed on remained in their posts for a few years before being forced to retire.

Others, such as Landau, died in retirement. Otto Blumenthal remained an editor of the leading journal * Mathematische Annalen * up to 1939. He died in Theresienstadt camp in 1944. Robert Remak emigrated to Holland but was later sent to Auschwitz, where he died. Felix Hausdorff committed suicide in 1943 when he was due to be transported. Although many outstanding mathematicians left, others, such as the great number theorist Helmut Hasse, stayed.

Segal gives considerable attention to the situation in the years leading up to the Nazi takeover, when German mathematics was foremost in the world.

There were, however, already problems and internal tensions. There was division over how to respond to the boycott imposed on German as well as Austrian, Hungarian and Bulgarian mathematicians after the Great War. In 1926, the boycott was lifted, and the Germans were invited to the 1928 International Congress in Bologna. Mathematicians of a nationalist bent proposed a counter-boycott, and the Prussian Academy rejected the invitation to send official representatives to Bologna.

Another point of controversy was on the foundations of mathematics. Here, the split was on the same personal lines as the boycott question. L. E. J.

Brouwer, who was Dutch but was sympathetic to German nationalism, took the lead in questioning the foundations of mathematics and was influential in rejecting the Bologna invitation. David Hilbert, who led the effort to show that mathematics could be completely axiomatised, was the leader of the internationalists and wrote to all rectors of German universities urging acceptance in "the interest of German science and respect".

There was also a wider academic crisis arising from the position of the professors as civil servants and their response to the Weimar state's claim to their loyalty. This and many other background issues are examined in the first three chapters of this book. The first, titled "Why mathematics?", discusses the special qualities of mathematics as a supposedly culture-independent abstract zone of reasoning amid the highly charged atmosphere of Nazism; the second deals with the foundational "Crisis in mathematics"; while the third covers "The German academic crisis".

The response of the mathematicians when the Nazis came to power ranged from risky interventions on behalf of threatened colleagues to enthusiastic support for the regime. Interestingly, no mathematician seems to have played any part in the resistance.

People today emphasise the importance of mathematics for the scientific, engineering and economic strength of society. Under the Nazis, the supposed racial and biological basis of mathematics was emphasised. New biologically based arguments had to be found. For example, the Jena mathematician Robert Koenig wrote in 1940 how number theory leads to a "biological drive to a new adaptation, a new means of orientation must be found, the complex number". The Nazis regarded the more mathematical aspects of physics such as quantum mechanics and relativity as "Jewish physics", so there was no credit to be had in that direction.

A recurring theme among mathematicians is the idea that there are specific national styles in mathematics. This idea reached its height in the founding of the journal * Deutsche Mathematik * by the outstanding mathematician Ludwig Bieberbach in 1936. This contained pedagogical articles and book reviews as well as research. There were articles stressing the mutual influence of mathematics and character, and in one of them it was claimed by the national leader of mathematics students that thereby the "image of the mathematician as a laughable figure of scorn will be overcome".

The Nazis had no particular interest in mathematics, pure or applied. As Segal shows, the subject's popularity declined in the new climate. In Munster in pre-Nazi times, beginners' lectures in mathematics drew 200 students on average, but only 50 in 1933. Nonetheless, mathematics remained attractive to the more advanced students, so that professors could enjoy an inner immigration and devote themselves all the better to their research.

In this environment, Heinrich Behnke, who incidentally was a widower with a Jewish son to protect, was able to found a school of several complex variables, which was to become the spur for the postwar revival of German mathematics under the leadership of Friedrich Hirzebruch.

Segal's fascinating study contains brief biographies of 16 representative mathematicians and a chapter on three case studies of controversies. Other chapters explore academic mathematical life and the mathematical institutions. There is a special chapter on * Deutsche Mathematik * . The author was denied access, in 1988, to some archives of the German Mathematical Society while archival work was being done. These archives were opened in 1996. The book includes an invaluable index of names, many with dates of birth and death but, unfortunately, no general index. It also contains detailed footnotes, an extensive bibliography, a list of archival sources and a list of 17 interviews carried out by the author.

* Mathematicians under the Nazis * will be of immense interest to both mathematicians and general readers, and it also provides a valuable insight into a traumatic period.

David Simms is an emeritus fellow of Trinity College, Dublin.

## Mathematicians under the Nazis

Author - Sanford L. Segal

Publisher - Princeton University Press

Pages - 530

Price - £55.00

ISBN - 0 691 00451 X