Prime case of numbers indivisible

It makes me very happy that after a long hard time all of the mathematicians of the world are represented here. This is as it should be and as it must be for the prosperity of our beloved science... It is a complete misunderstanding of our science to construct differences according to peoples and races, and the reasons for which this has been done are very shabby ones. Mathematics knows no races... For mathematics, the whole cultural world is a single country." The speaker, who received a standing ovation for this forthright affirmation of the scientific ethos, was the great German mathematician David Hilbert. The occasion was the Eighth International Congress of Mathematicians at Bologna in 1928, the first post-war congress to which German mathematicians had been invited.

The tortuous politics behind this event and, more generally, the inter-relation between politics and science, is the main theme of this account by Olli Lehto of the history of the International Mathematical Union. It covers in essence the whole of the 20th century, and, by focusing on the role of key individuals, it brings to life what might otherwise be a dry academic study.

Formal international cooperation in mathematics started at the end of the 19th century and culminated in the First International Congress in Zurich in 1897. The major role of France and Germany was reflected in the fact that the second congress took place in Paris in 1900 and the third in Heidelberg in 1904. The Paris congress holds a special place in the history of mathematics because it was there that Hilbert set out his famous list of problems for the century - the possible repetition of which in the year 2000 both fascinates and intimidates the world community.

The world war of 1914-18 put an end to these four-yearly congresses, and its aftermath left a legacy of hostility that poisoned the air in political and scientific circles. The International Research Council, covering the whole of science and giving birth to specialised bodies such as the IMU, was set up in the immediate postwar years, and it specifically excluded Germany and its allies. The IMU and the closely associated international congresses were equally affected, and this continued for a whole decade.

The main opposition to German mathematicians came, as might be expected, from France. A country where intellectuals feature prominently in public life was inevitably going to find it more difficult to disentangle politics from science. It is remarkable how many eminent French mathematicians of the period also held high public office. Paul Painleve was prime minister, Emile Borel minister for the navy, Raymond Poincare (cousin of the mathematician Henri Poincare) was president of the republic. The dominant figure, and the most hostile to German mathematicians, was Emile Picard. He was president of the International Research Council from 1919 to 1931, honorary president of the IMU (1920-32) and permanent secretary of the French Academy of Sciences from 1916.

The French position did not go unopposed, particularly in Britain, the United States and Scandinavia. G. H. Hardy, the leading British mathematician, was an early and outspoken opponent, as might have been expected from a staunch supporter of Bertrand Russell's pacifist stand and his subsequent troubles with Trinity College, Cambridge. The Americans, who were becoming increasingly important, had little sympathy with the French attitude. Eventually Picard had to concede defeat, and the Bologna Congress, presided over by Salvatore Pincherle, was the turning point. The IMU itself, as opposed to the congress, was not so fortunate. Its internal politics led to its demise in 1932, despite heroic efforts by its last president, the British mathematician William Henry Young.

The lessons of history are sometimes heeded. The political mistakes of 1918 were not repeated in 1945. Germany was not ostracised, and this applied also in the scientific sphere. The US was now the dominant power, and American mathematicians under Marshall Stone took the lead in organising the postwar congress at Harvard in 1950 and in setting up a new IMU. The statutes of the IMU were clear and explicit: no country was barred. Even in France, attitudes in 1945 were different from 1918. The leading French mathematician Henri Cartan (a future president of IMU), whose family had suffered grievously in the war, was one of the first to visit Germany to meet and assist his mathematical colleagues.

If the French-German problem had been resolved, the second half of the 20th century saw other political divisions, and the IMU had to struggle with these. For many years Russia proved a difficult partner. This was particularly unfortunate because Russia had a great and continuing mathematical tradition. In the early days post-1945 few Russians were allowed abroad to attend conferences and those who went were carefully selected by the political hierarchy.

Germany's East/West division also caused problems, as did the question of China and Taiwan. These mirrored similar problems at the United Nations. One might have thought that scientists could solve their problems of cooperation more easily than the politicians. For mathematicians, at least, the opposite was usually the case. The China/Taiwan issue is instructive. At the UN, realpolitik eventually won the day, but the IMU had the luxury of moral scruples. Although its statutes had been carefully drafted so as not to refer to nation states, it took nearly 30 years to devise a formula acceptable to the IMU, China and Taiwan. The combination of mathematical precision and oriental subtlety was a hard nut to crack.

A unique difficulty faced the IMU at the time of the planned Warsaw congress of 1982. Martial law had been declared and the country was in turmoil. Many in the West felt that the congress should be cancelled as a political protest. But the Polish mathematicians who had worked so hard in preparation, and who were not responsible for the actions of the communist government, were understandably keen to hold the congress. Eventually it was decided to postpone the congress by one year. It was successfully held in 1983, by which time the political situation was improving.

There are those who believe, honestly but perhaps naively, that science should and can be insulated from international politics. In reality this is not always easy, and genuine ethical dilemmas occur. For instance the Japanese were planning to host the 1990 congress at a time when UN sanctions against South Africa might have prevented South African mathematicians from attending. The problem was solved, with honour on all sides, but it was a close call.

Lehto, as secretary of the IMU for seven years and as organiser of the Helsinki congress in 1978, has recorded an eventful century of international mathematical organisation and has enlightened the official records with sensitive accounts of what went on behind the scenes. While primarily addressed to the mathematical community, who will be familiar with the main actors, the book should attract a wider audience interested in the way in which politics and science interact.

Sir Michael Atiyah, OM, was formerly master, Trinity College, Cambridge.