Once again, and in more than one respect, this voluminous book raises the question of how the history of mathematics can or should be written, and for whom?
Very wisely, the author chose to call it a book on "mathematicians", not on mathematics. While arguably the emigration of Jewish mathematicians from Germany and German-occupied Europe after 1933, and the war with its military applications of mathematics, were the two most important effects of National Socialism (henceforth NS) for mathematics as a whole, the American mathematician Segal deliberately excludes the first theme "except as this is important to other topics" (p.xv) and does not say much on the second one.
Not requiring "any advanced knowledge about mathematics" (xi), the book's foremost intent is to oppose the "general but perverse social view of mathematicians as disembodied intellects" (xi). Certainly no period in history seems more appropriate to be considered in this respect than Nazi rule in Germany, when neither the most apolitical and aloof stance nor the most opportunistic attitude would have helped a mathematician get along with the regime, or even to survive, if he/she happened to be a Jew.
But how about the behavior of the German mathematicians remaining in academic positions in Germany? In fact, the "mathematicians" discussed in the book are for the most part non-Jewish: there were no others who could act at that period of time. The fundamental opportunism of many, and a potential source of guilt feelings among them, hinge on the circumstance that they were not affected, that they were in this sense privileged, sometimes even the beneficiaries of the expulsion of their colleagues. Segal devotes the whole Chapter Eight "Germans and Jews" (pp. 419-492) to this relation and the connected psychology, not systematically, but from the perspective of individual biographies.
The author rightly opposes the "naïve idea that mathematicians 'analytically trained' ...ought not to have been prone to Nazi-fellow traveling" (xii). Along this line, he discusses the actions of Bieberbach, Teichmüller, Vahlen etc. The racist pseudo-theories of Deutsche Mathematik are the best known (and infamous) example. The treatment in chapter seven (pp. 334-418) sums up much of the available historical work on this aspect of the story.
Segal discusses at length (Chapter Six: Mathematical Institutions..., pp.229-333) the stimuli for collaboration and opportunism coming from such professional interests as maintaining a mathematical society (DMV or Mathematischer Reichsverband or GAMM), doing war research in mathematics, and striving for the establishment of a Mathematical Research Institute (Oberwolfach, founded in 1944). Because these professional interests concerned both big names such as Bieberbach and Hasse and minor mathematicians such as E. A. Weiss, M. Steck, U. Wegner, E. Tornier, E. Weinel and R. Weyrich, the author gives the latter as much attention as the former. In fact the sections on the "mathematical camps" for students organized by E. A. Weiss (pp.188-197) and on "Mathematics in the Concentration Camps" (pp.321-333), where the forgotten mathematician K.-H. Boseck was instrumental, are among the most richly documented parts of the book. The discussion of M. Steck's edition of J. H. Lambert's work in cooperation with the German Research Council and Swiss mathematicians late in the war (1943) is also new and interesting (pp.244-253).
This leads to the question of the originality of Segal's book, the sources used and its relation to work by others.
With his biographical study (1980) on the political role of the famous number theorist Helmut Hasse, Segal was among the first scholars to point to the parallelism of strong nationalism and outstanding research in some German mathematicians during NS. Almost the entire literature on mathematics and mathematicians under NS which followed was written in German. No complete "history of mathematics during NS," which would include the totality of institutional, political and cognitive developments, has been published so far. German monographs on singular and relevant aspects of that topic such as modernism and mathematics (Mehrtens 1990), mathematical reviewing in Hitler's Germany (Siegmund-Schultze 1993), and emigration (Siegmund-Schultze 1998) do exist. These books are not mentioned in the bibliography or in the preface of the present book (although one of them appears in a footnote), apparently because Segal is predominantly aiming at a biographical perspective. Several articles by the same authors are also lacking, some of them in English (most notably Mehrtens's 1989 article on the DMV), and likewise papers by authors such as Th. Hochkirchen (on Tornier), V. Peckhaus (on H. Scholz), and V. Remmert (on W. Süss and G. Doetsch). The bibliography does not contain titles which appeared 1999 or later. The fact that Segal also does not cite the original 1992 publication of his own interesting study on E. A. Weiss and the mathematical camps might lead to the conjecture that the author is just not concerned about bibliographical completeness.
A similar remark concerns the archival sources. Segal makes extensive use of the NSDAP-files in the former Berlin Document Center. In this respect he adds to the literature. On the other hand, discussion of other sources, such as the F. Kubach papers (Washington/Koblenz), and, above all, the DMV-Archives in Freiburg, is notably absent. The latter were opened to research in 1997, which was, according to Segal "not in time to be used in the composition of this book" (p.264). The reliability of the references to the sources as they stand in the book is generally good. Some minor mistakes (such as p. 204, the alleged emigration of Th. v. Kármán from Germany after 1933 — he in fact went to Pasadena already in 1930) are tolerable. The careless treatment of German orthography in the bibliography and even in the displayed quote from Thomas Mann (p.vii) is lamentable. The photographs of central figures in the middle of the book (after p.228), which are almost all from the Oberwolfach Institute Collection, do not do justice to the quality of the available originals (e.g., the swastika on Vahlen's jacket is not recognizable).
The book should be most useful to readers who are either mathematicians who want to know something about the behavior of their colleagues in a difficult time or to academics from other fields who want to get — via individual biographies — some first notion of how life was like in Nazi Germany. It will be less valuable for professional historians. Still, the book brings one particular aspect of the literature (mostly in German) on the subject to the attention of a broader English-reading public, and also presents some new research by the author himself. Together, these make it a valuable contribution. One hopes that it will create further interest in an important topic in the history of mathematics.
 The work comes closest to mathematics itself in the appendix, which gives information on the number of articles in the main fields of mathematics that were published in the four leading German mathematical journals between 1933 and 1944. This appendix (pp.493-507) was compiled by Segal's student Beata Smarczynska.
 Some exceptions are Landau, Courant, Rado, Blumenthal, and Hausdorff, who were marginalized, expelled, killed or driven to suicide.
Reinhard Siegmund-Schultze (Reinhard.Siegmund-Schultze@hia.no) is a German historian of mathematics, born in Halle 1953. He is currently professor at Agder University College in Kristiansand (Norway). His publications include books (in German) on mathematical reviewing (1993) and on the emigration of mathematicians during NS (1998) as well as (in English) on the support of mathematics by the Rockefeller Foundation (2001).