The Nobel of Numbers

John Charles Fields is a little-known Canadian. He deserves to be better known. Turbulent Times in Mathematics: The Life of J.C. Fields and the History of the Fields Medal, a fine biography and account of his career and work by Elaine McKinnon Riehm and Frances Hoffman, may help to rectify that situation.

Fields is best known today among the world’s mathematicians for having established an award that has taken the place of a Nobel Prize for mathematics. No one really knows why Nobel overlooked mathematics when he set up the prizes in his will. There are rumours, but no real evidence as to Nobel’s reason. Fields, through his prominence in North America and Europe, saw the need and opportunity and set out to fill it. The medal he created, officially known as the International Medal for Outstanding Discoveries in Mathematics, was to be awarded every four years, to two, three or four mathematicians not over the age of 40, at the quadrennial conference of the International Mathematical Union. Fields died before the medal was first awarded: it immediately became known as the Fields Medal.

Fields was born in 1863 and raised in Hamilton. He benefited from excellent schools that had been established in Hamilton, undoubtedly under the influence of Egerton Ryerson.

During the years when Fields was a student, graduates of the Hamilton Collegiate took first or second place in classics and mathematics at the University of Toronto every year. (Would that our high schools still taught classics instead of some of the intellectually weak content now included.)

Fields graduated from the University of Toronto in 1884 with the gold medal in mathematics. He then went to Johns Hopkins University in Baltimore to pursue a PhD. Johns Hopkins had been founded only in 1876, based on the German model, with a commitment to research and offering a doctorate. No Canadian university then offered such a degree. (Neither Oxford nor Cambridge offered a PhD until the 1920s, and only then because foreign graduate students were all going to Germany to get them.) Fields enjoyed the atmosphere at Johns Hopkins, spent two more years there, and then taught for three years in a college in Pennsylvania.

He went to Europe in 1892, spending two years in France, followed by five in Germany in Berlin and Göttingen. He became fluent in French and German. And he became well acquainted with the leading mathematicians in Europe, also visiting Britain and Scandinavia.

The community of scientists was small, and while their meetings tended to be formal with long speeches, they also enjoyed casual evenings with beer in taverns.

After a year at the University of Chicago, Fields returned to Canada in 1901, taking up an appointment in mathematics at the University of Toronto, and established a pattern of teaching at Toronto during the academic year and travelling to Europe for the summer months. He followed this pattern for the rest of his life, except for the war years. His biographers estimate that, over the years, he spent more than 50 weeks criss-crossing the Atlantic. How did he manage this on a modest professorial income? He evidently lived quite simply in Toronto, “rooming” in one of the substantial family homes in the neighbourhood just to the west of the university. He never married, never owned a home. He had had a small inheritance but was never a wealthy man.

He had become convinced from the examples he had seen at Johns Hopkins and in Berlin and Chicago that universities and their professors should be committed to research. But it was not until 1915 that one of his students, Samuel Beatty, received the first PhD in mathematics awarded in Canada. Fields also believed that scientific research should be a concern of government, business and the public. Not only as an intellectual pursuit, “for the sake of knowledge,” but also as a basis for economic advancement and competitiveness. He lobbied Queen’s Park and Ottawa intensively as well as his friends in business. He was involved with, and led for some years, the Royal Canadian Institute, which offered lectures on science on Saturday nights at Convocation Hall at the University of Toronto. These were popular and well attended for many years.

He hoped that there might be philanthropic support for research in Canada such as had been provided at Johns Hopkins (by Hopkins himself) and at Chicago (by the Rockefellers). But it was to be almost one hundred years before Mike Lazaridis founded and endowed the Perimeter Institute for Theoretical Physics as a major centre for research and the training of researchers. Fraser Mustard had established the Canadian Institute for Advanced Research in 1982 with the hope of major philanthropic support, but it never arose.

Fields’s own research, including his doctoral work and the work he had completed in Berlin, provided the basis for a book, Theory of the Algebraic Functions of a Complex Variable, which was published in Europe in 1906. This and other papers that he had published were highly regarded and led to his recognition as “a mathematician of the first rank.” They led to his being elected a fellow of the Royal Society of Canada in 1909 and a fellow of the Royal Society of London in 1913. His book was the first publication of an important mathematical monograph by a Canadian. Yet his work, highly regarded at the time, was subsequently overshadowed by modern approaches that used abstract algebra.

The last decade of the 19th century and the early years of the 20th were golden years in Europe and golden years for international science. Fields’s years at universities in Europe and his close acquaintanceship and friendship with many leading European mathematicians were a great source of pleasure to him. And he was profoundly affected by the spirit of the international organizations that had arisen to facilitate and foster the meetings of scientists that became powerful stimuli to advancement. He believed strongly in the international culture of science.

Fields attended the first International Congress of Mathematicians convened at Cambridge in England in 1912. But that hopeful beginning was quickly stymied. The outbreak of war in 1914 led to a deep schism among European scientists, particularly between the Germans and the French and their allies. This was to last for many years. What is amazing is that during the no less bitter Napoleonic Wars British and French scientists regularly and peacefully visited back and forth between London and Paris to present scientific papers and be entertained at their respective academies.

When the time came in 1920 for the first post-war International Congress of Mathematicians, under French influence and with great symbolic significance, it was held in Strasbourg. This was, of course, in Alsace, which France had lost to Germany in 1871 and only recovered with the sorting out of borders in 1919 after the First World War. The Germans and their allies were excluded from the Congress, to Fields’s regret and to that of other eminent scholars including G.H. Hardy, England’s leading mathematician and the author of the classic A Mathematician’s Apology.

It had been expected that the next international mathematical congress would be held in the United States. But disagreements and controversy in America destroyed support for the congress. Fields saw his opportunity and, utilizing his connections in Ottawa and in Toronto, generated support for an invitation to hold the 1924 congress in Toronto. In spite of Canada’s weakness in mathematics and scientific research generally, the 1924 congress proceeded and was a great success. It concluded with a 17-day rail excursion to the Pacific coast and back. It was all Fields’s triumph.

At the congress, the Americans proposed that the policies that had excluded the Germans and their allies be changed to readmit them. It took four years of politicking in Europe, in which Fields played a significant role, for the policy to be reversed. Finally, at the 1928 congress in Bologna, the Germans participated. It had taken a decade since the end of the war.

It was after the 1928 congress that Fields, perceiving that “the rift is still huge,” developed his idea for an international award in mathematics that in his opinion would help repair it. He started speaking about his idea in Europe and in North America. He found support among national mathematical associations. Fields developed his plans carefully. He was determined that the medal should honour work done by mathematicians under the age of 40. The medal was designed by the Canadian sculptor R. Tait McKenzie. It has a figure of Archimedes on its front and a classical Latin quotation on the reverse, which translates as “mathematicians gathered together from around the world honour noteworthy contributions to knowledge.” The recipient’s name would appear on the edge. Fields’s name does not appear. He died in 1932, committing most of his fairly modest estate to establishing the prize.

The medals were first awarded in 1936. With a break until the 1950 congress in Cambridge, Massachusetts, but with none of the animosities that characterized the years of and after the First World War, 52 medals have since been awarded at 17 congresses.

The authors of this book have done a superb job with exhaustive attention to the details of the history, both of Fields’s efforts and of the mathematical organizations. The only conspicuous error suggests that Bertrand Russell lost his position at Oxford because of his pacifist opposition to the First World War: it was Trinity College Cambridge that had dismissed him.

Fields would be pleased and probably surprised to know how mathematics has developed in Canada since his time. It used to be a subject chosen for its intellectual challenge, but offering careers primarily in teaching and, occasionally, in actuarial work. Mathematics is still valued for its intellectual rigour, but now offers many career prospects in the modern economy. There are good departments of mathematics in many universities across Canada. The University of Toronto remains strong. But led by the migration of a number of mathematicians from Toronto to Waterloo, the University of Waterloo has developed a concentration in mathematics. Waterloo now has the largest enrollment in mathematics in North America, and probably in the world. Toronto and Waterloo enjoy a healthy rivalry, not only between themselves, but with the best in the United States.

In 1992, mathematics professors from McMaster University and the universities of Toronto and Waterloo conceived of a centre for research that would have no permanent academic staff, but would bring researchers from across Canada and abroad to work together for a period on a defined area. They were successful, and the Fields Institute was established on the Toronto campus, where it occupies a purpose-built home.

The American Mathematical Society conducts annual contests, known as the Putnams, for university students. Against teams from Harvard, MIT and Caltech, in recent years, teams from Toronto and Waterloo have each ranked in the top five 18 times. A remarkable number of young Canadians now choose to study mathematics. This, as well as the medal and the institute, can be seen as Fields’s legacy.