The Young Fields

John Charles Fields, always known as J.C. Fields, was born in 1863 in Hamilton, Ontario, then Canada West. His father, after whom he was named, was a tanner whose small leather shop in the city dealt in hides and leather- working tools. His mother, Harriet Bowes Fields, had been a school teacher before her marriage. Fields received his education in Hamilton public schools, first at Central School and then at Hamilton Collegiate. In the 1860s and 1870s both schools were famous for their teachers, curriculum, and the success of their graduates. Fields won the gold medal in mathematics in his final year. He graduated from University of Toronto in 1884, again winning the gold medal in mathematics, and proceeded directly to graduate school at Johns Hopkins University.

The mathematics department at Johns Hopkins had recently been established by English mathematician James Sylvester (1814-1897). During his years at Hopkins, Sylvester introduced the "seminary" into mathematics teaching in North America and established the American Journal of Mathematics. He returned to England in 1883, the year before Fields arrived, but he left behind a strong, young faculty that included Thomas Craig, from whom Fields took many courses, and who is presumed to have been his thesis advisor. Fields received his PhD in 1887 for a thesis entitled Symbolic Finite Solutions and Solutions by Definite Integrals of the Equation dny/dxn = xmy.

Encouraged by Craig, Fields joined the Société mathématique de France (SMF) while still a graduate student, a membership he kept for life. After two further years as teaching fellow at Johns Hopkins, he joined the faculty of Allegheny College, Meadville, Pennsylvania, where he taught until 1893. By that time he had published several papers, including one in the Journal für die reine und angewandte Mathematik ("Crelle").

At Allegheny College, Fields became convinced that his interests lay in research. With a small inheritance from his parents, who had both died when he was still at home in Hamilton, and with an introduction from Craig, Fields went to Paris where he spent almost two years taking courses at the Collège de France and the Sorbonne. He published nothing during those years, but attended the bi-weekly meetings of the SMF where he heard lectures by Henri Poincaré, Émile Picard, Paul Painlevé, and Gabriel Koenigs, among others. He learned to speak French. He also made acquaintances among French mathematicians, particularly Koenigs, which he renewed frequently in the years that followed.

In 1895, he moved to Germany, spending six months in Göttingen, where he attended courses given by Felix Klein. He then went to the University of Berlin where he studied for the following six years. Fields lived an intense mathematical life in Berlin, attending many courses in mathematics and several in physics, chemistry, and philosophy. His experience in Berlin affected him profoundly, in particular the emphasis by German universities on research. He has left 111 notebooks, now in the University of Toronto Archives, from courses he attended or notes copied from other students. These have been studied in detail by Marcus Emmanuel Barnes in an MA thesis entitled "John Charles Fields: A Sketch of His Life and Mathematical Work." In Berlin, Fields fell under the influence of Karl Weierstrass (1815-1897) through courses on his work given by Herman Amandus Schwarz (1843-1921). Of Weierstrass, Fields wrote: "He was the incarnation of rigor in his mathematical methods. He demanded perfect precision in the formulation of a problem and absolute rigor in its demonstration." Fields published no papers during his Berlin years, but completed an early version of his book, Theory of Algebraic Functions of a Complex Variable (1906). His 356-page manuscript for the book, likely dating from 1898, is in the Gerstein Science Library at University of Toronto. Although the fountain pen had been invented, Fields wrote in ink using a straight pen. He dips his pen, writes a few lines, the ink fades, and he dips again. He later took to the typewriter. While in Berlin, Fields learned German and also met many young German and American mathematicians studying there.

Fields' stay in Europe made him at home in both French and German, and gave him the beginning of a wide circle of scientific acquaintances that he added to in his future travels.

Return to Canada

Fields returned to Canada in 1901 to give a course at the University of Toronto on Hilbert's work and was appointed Special Lecturer in 1902. He remained on the faculty until his death in 1932. He became a Fellow of the Royal Society of Canada in 1909 and of the Royal Society, London, in 1913, and was made Research Professor in 1923. In Toronto, over a period of three decades Fields promoted the research ideal he had come to know at Johns Hopkins and the University of Berlin. He did this within the university and beyond. Shortly after his return to Canada, he joined the Canadian Institute, renamed the Royal Canadian Institute (RCI) in 1914, serving as President from 1919 to 1925. Like others before him, Fields tried to make the Institute a vehicle for active research not merely a forum for encouraging public interest in science and disseminating results. Like the others, he failed. The RCI, founded in 1849 by engineer Sandford Fleming, remains active today, however. Its public activities touching on a wide variety of scientific topics continue to draw a large audience, including those on mathematics sponsored by the Fields Institute.

Fields also published several non-mathematical articles on various subjects-German university mathematics, science and industry, avoiding "brain waste," and in praise of John D. Rockefeller. He was always on the look-out for a Canadian counterpart to Rockefeller or Andrew Carnegie who might provide money to support scientific research. Fields' constant work to promote the culture of science in Canada made him a well-known figure outside the university.

In the years before 1914, Fields went to Europe most summers. Travel was his one extravagance in an otherwise "abstemious" and certainly non-materialistic life. He attended the great Abel Centenary in Norway in 1902 in addition to numerous meetings of the British Association for the Advancement of Science (BAAS) and the American Association for the Advancement of Science (AAAS). At the BAAS meeting in Glasgow in 1901, he met Swedish mathematician Gösta Mittag-Leffler (1846-1927), who encouraged Fields to publish his book, suggesting the Berlin publishing firm of Mayer and Müller. Mittag-Leffler's friendship was important to Fields, who admired the older mathematician's cosmopolitanism and internationalism. It was likely at Mittag-Leffler's suggestion that Fields was appointed to the organizing committee of the 1912 Cambridge International Congress of Mathematicians, the first Congress Fields is known to have attended.

World War I and Science

Before 1914, mathematicians had considered mathematics to be the most international of the sciences, free from "sub-turbulent" politics. But the war changed all that. A month before the Armistice of November 1918, a group of scientists from the scientific academies of the allied countries known as the Entente cordiale (Belgium, Britain, France, Italy, Serbia) plus the United States met in London to consider the nature of scientific relations to be resumed at war's end. They declared their intent not "to resume personal relations in scientific matters with their enemies until the Central Powers" (Austria, Germany, Hungary, and Turkey) could be "re-admitted into the concert of civilized nations." This declaration was reaffirmed by the International Research Council (IRC) established the following summer in Brussels. Under the IRC, the various sciences were encouraged to form unions from which the scientists of the Central Powers were excluded. Provisions were made, however, for representation from nations that had been neutral during the war.

When mathematicians founded the International Mathematics Union (IMU) at the Strasbourg Congress (1920), they did so under the rules of the IRC, thus excluding mathematicians from the Central Powers. In this way, the dividing lines of World War I continued after the war in mathematics.

At Strasbourg, American mathematician L.E. Dickson (1874-1954) of the University of Chicago issued an invitation on behalf of the American Section of the IMU to hold the 1924 Congress in the United States. It seemed to everyone like a good idea to move it across the Atlantic and out of the war-ridden atmosphere of Europe. Dickson gave the invitation in the hope that by 1924 war feeling would have subsided and it would be possible to invite all mathematicians of the world to attend. But, he underestimated the depth of feeling especially among French mathematicians. As the United States had been one of the founding members of the IRC and a signatory to its exclusion policy, this was a problem. The American Mathematical Society (AMS) was badly divided between those wanting to honour the invitation notwithstanding the exclusion rules and those who found the idea abhorrent.

International Mathematical Congress, Toronto 1924

In late December 1921 at a large meeting in Toronto of the AAAS, which Fields had organized splendidly, a casual suggestion was made to hold the 1924 Congress in Toronto. Whereas, up to 1924, nations had vied with one another for the honour of holding the Congress, in 1924 nobody wanted it- except Fields. He saw the Congress as an opportunity to boost mathematics in Canada, which he felt lagged behind physics. Negotiations followed, and the IMU transferred the 1924 Congress to Toronto. At the time, Fields had no money, and there was as yet no Canadian Mathematical Association to help with organization. (The CMS was not founded until 1945.) He mounted a strenuous and successful campaign to raise money for the Congress from the provincial and federal governments. Then, with money in hand to pay the travel costs of delegates from abroad, he hurried off to Europe to promote attendance. At home, on the practical side of the Congress, he had the help of a strong organizing committee at the University of Toronto, chaired by a friend, physicist John C. McLennan. He also had the help of two friends in the mathematics department, the young Irish applied mathematician J.L. Synge (1897-1995) and French geometer Jacques Chapelon (1884-1973). Both men, known for their wit and sense of humour, added lightness to Fields when his responsibilities were heavy. However, the entire university also pitched in to help.

The 1924 Toronto Congress was a success-it was attended by 444 mathematicians from 27 countries. Fields, now 61, was exhausted by his work on the Congress and by the trans-continental train excursion that followed. He developed heart problems, and was never robust again.

According to the rules of the IMU, Germany, Austria, Hungary, and Turkey were excluded from the 1924 Congress. But the issue of their inclusion at subsequent congresses was raised in a motion by the American delegates. Their resolution asked "if the time was ripe for the removal of the restrictions on membership now imposed by the rules of the [International Research] Council." It was supported by Britain, Denmark, Italy, Holland, Norway, Sweden, and the United States. Not Canada. We do not know how Fields felt personally about the exclusion. But we have an inkling. Up to 1912, every congress was known as the "International Congress of Mathematicians." After the exclusion, which he vigorously opposed, Mittag-Leffler succeeded in having the IMU accept a small change in wording to "International Mathematical Congress." This was intended to indicate that although the mathematics might be international, the mathematicians attending were not. The French ignored this in 1920 in Strasbourg, but Fields observed it. The Toronto Congress is the only one so named. We also know that throughout his life including the period from 1914 to 1918, he spoke and wrote in praise of the German university system. He was always careful to distinguish between education and politics.

Shortly after 1925 when Germany was admitted to the League of Nations, German mathematicians were then invited to join the IMU. They declined, however, to be a part of an organization that had recently shunned them. This was the situation in 1928 when Salvatore Pincherle (1853-1936) was planning the Congress in Bologna. Pincherle sidestepped the rules of the IMU by inviting mathematicians from all countries to the Bologna Congress. In order to convince the Germans to attend, he declared the Congress to be under the aegis of the University of Bologna, not the IMU. The moment when the German delegation entered the hall, led by David Hilbert, has often been described. The audience rose in a standing ovation. Hilbert made a brief remark that "all limits, especially national ones, are contrary to the nature of mathematics."