This book is not a history of the mathematics of the twentieth century. Instead, it is a history of the international mathematical community of the twentieth century. As the end of the century draws near, it is fitting that we take a look back at the development of international cooperation in mathematics over the past decades. Olli Lehto carefully lays out this history with rich detail and abundant references to material contained in the International Mathematical Union (IMU) archives at the University of Helsinki. In his preface, Lehto gives his own reasons for writing the book:

The question may be asked why this history of the IMU has been written. The classical answer is to express hope that some lessons might be drawn from it, that the record of past events might serve as a guide for the future. In historia semen futuri.
A more concrete justification for the creation of this book is the fact that the IMU is not particularly well known among mathematicians. (...) I hope that this history will improve understanding of the important role the Union has played in the promotion of mathematics throughout the world.

Finally, this is a story of how ideas of the global cultivation of mathematics, across national borders, gradually began to take shape a century ago and how these ideas developed, amidst political difficulties and serious setbacks, into a fruitful worldwide cooperative effort under the aegis of the IMU.

It is interesting (though certainly not surprising) to note the extent to which the history of international cooperation in mathematics in the twentieth century parallels the political history of the same period. The book begins with some prehistory; a description of the dawning of international cooperation in mathematics in the late 1800's. A systematic cataloging of all international mathematical publications developed in Germany and France in the 1870's and 1880's, and the first International Congress of Mathematics (ICM) was held in Zurich in 1897. The ICMs were held every four years from 1900 through 1936 with the exception of 1916 (due to the war). Meanwhile, the International Association of Academies (IAA) was formed in 1899 and met regularly through 1913. Though this association had no contact with mathematics, Lehto makes the case that it indirectly paved the way for the foundation of the IMU. World War I put an end to the activities of the IAA, and, after the war, the International Research Council (IRC) was constituted. One of the stated goals of the IRC was the formation of international associations or unions deemed to be useful to the progress of science. Thus, the first steps towards the formation of the IMU were taken in Brussels in 1919 at the Constitutive Assembly of the IRC. In accordance with this program, the IMU was formally founded during the ICM in Strasbourg in 1920.

The book goes on to describe the early years of the IMU as it struggled to take shape in the years from 1920 through 1932. There were political struggles as opposition grew to the IMU's policy of exclusion. The recently ended war made it politically difficult to bring together mathematicians from all countries, but some organizations - notably the American Mathematical Society - fought against the restrictions on participation in the Congresses (which were now being organized by the IMU). In 1928, the IMU separated from the Congresses and, in the fall of 1932, activities of the IMU were suspended entirely. They were not to resume until 1952.

The period from 1933 through 1939, leading up to World War II, was without an IMU, but some degree of mathematical cooperation continued. Fields Medals began being awarded in 1936, an ICM was held in Oslo in 1936, and international cooperation in mathematics education grew. World War II interrupted things briefly and the period from 1945 through 1951, just after the war, was a period of preparation for a "New" IMU. An ICM was held at Harvard in 1950, and preparation of statutes for a rebirth of the IMU were being prepared. In 1952, this New IMU was officially born with the First General Assembly in Rome.

I won't spoil things in this review by telling any more of the story. Lehto tells the full story and tells it well. He concludes the book with an appendix containing lists of things like Members of the IMU, General Assemblies of the IMU, Executive Committee information, International Congresses of Mathematics, Fields Medals, and the like. Also included are a lengthy set of endnotes and a very complete index.

Lehto has met his goal of writing a book which should make the role of the IMU more widely known and understood by mathematicians and other interested parties. He follows a large number of threads through the tapestry of twentieth century mathematics and gives us a tale of mathematical and historical interest. This book would be of interest to people ranging from university students of mathematics or history to established mathematicians who have lived through a larger piece of the century. It would fit nicely in the mathematics collection of any library.

Carl D. Mueller is Associate Professor of Mathematics at Georgia Southwestern State University in Americus, GA.