The author, as a graduate student, was shocked by the fact that some mathematicians, could be supportive defenders of the Nazi regime as a mathematician. An Alexander von Humboldt Fellowship in 1988 allowed him to consult archives and interview people in Germany. The result was this book that appeared in 2003 and now, posthumously (Segal passed away in 2010), an unaltered paperback version is made available.
It was Segal's intention to write this book not only for mathematicians and it is not meant to be a history of the mathematics developed during the Nazi period either. So there is no focus on the mathematics anywhere. However to understand the tension between personalities and their individual and sometimes quite different views on mathematics, the reader should be familiar with at least the general outline of what was happening on the mathematical scene in those days. At the same level, it is advisable that the reader has an idea about the German social and political situation and has some knowledge about the who-is-who in the period 1914-1946. Segal has a bantering, story-telling style, and it is sometimes difficult to keep track of the timeline and all the players involved if you do not have the general picture in mind.
The first three chapters are introduction. In the first Segal explains basically that mathematics is not the science of the cold indisputable truth, but that it is produced by humans with qualities and deficiencies. The second chapter is about the crisis in mathematics in that period. Early 20th century a revolution was started in algebra and analysis, but the very foundations of mathematics were shaken and mathematicians had to take a side, not only politically but also choose for example between Brouwer's intuitionism and the formalism of Hilbert and of course the protagonists on either side have their political preferences too. So the dispute could easily be confused with arguments about Aryan versus non-Aryan mathematics. In chapter 3, we learn about the academic crisis during the Weimar Republic (1919-1933). Jewish mathematical professors, like others colleagues, were dismissed allegedly because they were considered unfit because of their mathematical or pedagogical style. And thus became mathematics also politics and religion became a race.
Chapter 4 collects three case studies, described in great detail to illustrate the competition among bureaucrats but also between individuals. The first case is a conflict between Wilhelm Süss and Gustav Doetsch. Doetsch transformed himself in a 'full-blooded Nazi', while Süss, rector of the University of Freiburg, creator of the Oberwolfach institute, and president of the German mathematical Society (DMV) since 1938 was more moderate but nevertheless 'saved' the DMV by removing non-Aryan members. The conflict concerned a mathematics book project considered to have military importance in mid war. A second case is about the succession of Max Winckelmann, chair of applied mathematics in Jena in 1938. Finally Ernst Weinel was appointed but not before 1942 due to academic and political disputes. The appointment of Helmut Hasse in Göttingen is another case. Hasse got this strongly fought position, but he had to accept a 'true Nazi' Erhard Tornier as co-director, although the latter was quickly sidelined by Hasse.
A sketch of the academic life in general, but of mathematics in particular is the subject of the next chapter. How the regime did influence the organization of the universities but also the pedagogy and teaching. Sports camps were promoted rather than mathematics, and of course the staff was drastically reduced because of the war and ethnic and political cleansing. But of course also the number of students decreased, with a relative increase of female students as a consequence. Traveling and contacts with foreigners was limited. And there were specific Nazi visions on mathematics, not only at a university level, but also at secondary and elementary school level.
Next Segal discusses twelve cases of mathematical institutions. Institutions in a broad sense because it involves for example the composition of the editorial board and the publication policy of journals (like Mathematische Annalen), but also the board of the DMV and the role of Süss at a later stage, the Mathematische Reichsverband (which was an early adopter of the Führerprinzip), the creation of the Oberwolfach Institute in 1945, one of Süss's greatest achievements, and the applied mathematics organization (GAMM) founded by Prantl and von Mises. Most surprising is a section on the organized mathematical activity in concentration camps.
Chapter 6 about Ludwig Bieberbach is Segal's showpiece. Bieberbach and Vahlen were members of the NSDAP, were anti-Semite, and were founders of the Deutsche Mathematik (which in practice meant intuitionistic mathematics, a general Nazi ideology not restricted to mathematics alone). A detailed account is given of Bieberbach's 'Nazi career' and even gives a psychological analysis of why he grew so ardent an adept of the Nazi ideology in a relatively short period.
The last chapter is called `Germans and Jews'. This is again a list of different people, both Jews and nationalists. How they experienced the regime and how they reacted to it. Most of them may have preferred to be left alone with their mathematics, but they somehow had to interact nolens volens with Nazi politics and ideology. Some 16 mathematicians are considered among which Wilhelm Blaschke, Erich Hecke, Oswald Teichmüller, Richard Courant, Edmund Landau, Felix Hausdorff, Ernst Zermelo, and others.
History is not only made up of indisputable facts, but it is always a way to look back upon events from the past and interpreting them. Hence history is to some extent individual, and not all persons explain the same facts in the same way, partly because not all circumstantial and psychological elements and believes of all the actors are available. It can never be a full one-to-one account of the past. Hence different opinions will exist about historical interpretations. David E. Rowe gave a rather critical review of the 2003 edition of Segal's book in the The American Mathematical Monthly, 112, April 2005, pp.374-381. Some of this critique is not only a matter of interpretation. However, as also Rowe agrees, this book is still a massive source of information. When it comes to the amount of details and facts, Ludwig Bieberbach and Wilhelm Süss take the lead. Whether or not one agrees with all Segal's interpretations, the sources he consulted and the facts communicated is major achievement. The archives of the DMV were made public only in 1997, but Segal was in Germany at an earlier date, so that he could not consult them. Hence, although the original publication date is 2003, the research and perhaps the writing is probably older, and is not updated in this edition. So there are more recent papers and books available on the topic that one should consult. This book however will be a beacon that provides a solid foundation to build on.