Copenhagen 1871- Oslo 1948

Poul Heegaard (1871-1948), reproduced from (MUNKHOLM and MUNKHOLM 1999, p. 926).

Brief scientific biography

Poul Heegaard (1871-1948) was born 2 November 1871 in Copenhagen1. His father was a professor of philosophy at Københavns Universitet (the University of Copenhagen) with an interest in mathematics and the sciences. According to his later recollections, Heegaard's way into mathematics was indirect and highly curious. Because he had trouble with mental arithmetic but showed more aptitude for algebra, his teachers had given him more advanced algebra problems to solve - and he had felt the success of doing mathematics. In retrospect, he reflected that this aptitude "let my surroundings ... drive me towards an occupation with mathematics for which I did have a talent, but for which I do not have the burning interest that I have met in others" (MUNKHOLM and MUNKHOLM 1999, p. 927). Later in his mathematical career, his many other interests would often distract him from doing mathematical research, and his career as a research mathematician was based on less than a handful of truly original works. Instead, he spent much of his time lecturing on mathematics and the sciences to the public and on affairs related to the teaching of mathematics (to be described below).

Heegaard attended the gymnasium (Metropolitanskolen in Copenhagen) from which he obtained his artium in 1889. He then began studying at the University of Copenhagen, but not full-time because the death of his father had left him in a position where he had to support himself. He did so by tutoring other students and functioning as an examiner at the Polyteknisk Læreanstalt (Polytechnical College). While a student, Heegaard attended the lectures of the three dominant Danish mathematicians of the second half of the nineteenth century: H. G. Zeuthen (1839-1920), Julius Petersen (1839-1910), and T. N. Thiele (1838-1910). In 1893, Heegaard graduated after writing a thesis on algebraic curves in the tradition of Michel Chasles (1793-1880) under the supervision of Zeuthen.

After his graduation, Heegaard wanted to go abroad to continue his studies. Following the advice from the Francophile Zeuthen, Heegaard went to Paris in 1893 with letters of recommendation from Zeuthen. However, Heegaard's reception in Paris was cool, and he never had any really fruitful contacts with Paris mathematicians. In particular, it does not seem that Heegaard ever met Henri Poincaré (1854-1912) - either in Paris or later in life - although much of Heegaard's fame was connected to Poincaré's research (MUNKHOLM and MUNKHOLM 1999, p. 931). Frustrated with his situation in Paris, Heegaard decided to follow an earlier idea of his to go to Göttingen and study with Felix Klein (1849-1925). The stay with Klein proved to be more productive than Heegaard's time in Paris (MUNKHOLM and MUNKHOLM 1999, p. 931).

When he returned from his study tour in 1894, Heegaard began work on his dissertation while earning a living teaching mathematics at two or more gymnasia. The dissertation proceeded from discussions with Klein in Göttingen, and in 1897, Heegaard became aware of a recent publication by Poincaré on Analysis situs. In Poincaré's paper, Heegaard found that one of the central theorems (the duality theorem) was problematic, and he started searching for a counterexample. The analysis of Poincaré's duality theorem and Heegaard's counterexample made up the larger part of Heegaard's doctoral dissertation, which he defended in 1898 (HEEGAARD 1898). Its contents were also of great interest to the international community, and Heegaard communicated with Poincaré and others. A French translation of the dissertation was later published in (HEEGAARD 1916); for a description of the contents of Heegaard's dissertation, the reader is referred to (MUNKHOLM and MUNKHOLM 1999) and the references therein. Heegaard's quick rise to fame inspired the editors of the Encyklopädie der mathematischen Wissenschaften to ask him to write a chapter on the new analysis situs together with Max Dehn (1878-1952), resulting in (DEHN and HEEGAARD 1907). This collaboration initiated a lasting connection between Dehn and Scandinavian mathematicians.

After his return from his European tour and his graduation, Heegaard extended his teaching obligations to even more secondary and tertiary institutions in Denmark. Although these teaching jobs left him little or no time for continued mathematical research, Heegaard later remembered the period as peaceful and happy (see e.g., MUNKHOLM and MUNKHOLM 1999, p. 932). In particular, Heegaard was relieved from the tensions that he had felt in the group of Danish mathematicians at the University of Copenhagen and the Polytechnical College. Despite his relatively meagre list of mathematical research publications, Heegaard was appointed professor of mathematics at the University of Copenhagen in 1910 after a rather bizarre turn of events. Heegaard had been persuaded to apply for the vacant position after Zeuthen and there is evidence that he was reluctant to accept the position: it would mean a reduction in his salary (his salary at the University of Copenhagen would be only 1/5 of what he was earning teaching multiple jobs in gymnasia and other institutions; see MUNKHOLM and MUNKHOLM 1999, p. 936), he would have to focus more on research, and it would force him to deal with some of the tensions and conflicts that he had been happy to avoid previously. Despite his own reservations, Heegaard was appointed professor and held the professorship until 1917 when he suddenly retired claiming overwork and collegial problems as his reasons (see MUNKHOLM and MUNKHOLM 1999, pp. 935-936). Heegaard's strained relations with his colleagues in Copenhagen, in particular with Harald Bohr (1887-1951), played out both at the University and in the Danish Mathematisk Forening (Mathematical Society) until they were instrumental in bringing about Heegaard's resignation in 1917. The following year, Heegaard was called to a chair in geometry at Universitetet i Oslo (the University of Oslo) - the city was then still called Kristiania - and he left Denmark for a new career in Norway. Heegaard's resignation and subsequent move to Oslo made the headlines of Danish newspapers and tabloids. The character of Heegaard's conflicts with his colleagues in Copenhagen has been analysed in (RAMSKOV 1998, RAMSKOV 2004).

Contributions to education

From 1907 to 1911, Heegaard served as a consultant to the ministerial inspection of the secondary schools (undervisningsinspektionen) and he was teaching at a number of institutions in Denmark (see Johansson 1948, p. 38). In 1908, in connection with the fourth International Congress of Mathematicians (ICM) in Rome, the Internationale mathematische Unterrichtskommission (International Commission on Mathematical Instruction) was established and Heegaard was appointed the Danish delegate to the commission. He was thus, originally, a representative of the ministerial inspection and an experienced teacher in the gymnasium, and subsequently (after 1910) a representative of the academic mathematical milieu. In his capacity as a delegate to the International Commission on Mathematical Instruction, Heegaard wrote the first comprehensive report on the instruction of mathematics in Denmark (HEEGAARD 1912). In that report, Heegaard described the organisation of mathematics teaching in Denmark at all levels of education. In 1915, a foreigner's account of the Danish mathematical teaching was published as (ROHERBERG 1915).

After the initial setting up of the International Commission on Mathematical Instruction and Heegaard's appointment as delegate for Denmark, he took it upon himself to inform his Danish colleagues of the commission in the Matematisk Tidsskrift (Journal of Mathematics) by translating the commission's mandate (HEEGAARD 1909). A Danish sub-commission was formed headed by Heegaard and involving many of the centrally positioned mathematicians. Nevertheless, the work of the International Commission on Mathematical Instruction received only scant attention, and in 1912, Heegaard published a short notice in the Journal of Mathematics in which he drew attention to the work of the International Commission on Mathematical Instruction (HEEGAARD 1912). The notice begins by a story that perhaps illustrates how the International Commission on Mathematical Instruction was thought of by Danish mathematicians:
"A German recently told in a newspaper of how he had felt when a small child whom he had wanted to give an orange had declined the offer. The child was a Kriegskind [child of war] who had never seen an orange before. Similarly, many young mathematicians are likely to have grown up without knowing what is hidden behind the four letters that caption these lines [I.M.U.K.]" (HEEGAARD 1912, p. 108).

Poul Heegaard (1871-1948), reproduced from (Johansson 1948, p. 36).
Poul Heegaard (1871-1948),
reproduced from (Johansson 1948, p. 36).
Heegaard went on to describe how the proposed plans for a report from the International Commission on Mathematical Instruction at the planned ICM in Stockholm 1916 had been abandoned. Nevertheless, he wanted to draw the attention of "all those interested in these matters" to recent publications on the state and history of mathematics instruction in Germany and the United States (HEEGAARD 1912, p. 109). Similar attempts to draw attention to the work of the International Commission on Mathematical Instruction and its publications were also made in the Journal of Mathematics by e.g. (HECKSCHER 1912a, HECKSCHER 1912b, HECKSCHER 1913, I.M.U.K. 1914, TRIER 1915).

When Heegaard arrived in Oslo, he soon felt much more at ease and at home with the mathematical community there than he had done in Copenhagen. Soon, he organised the Norsk matematisk forening (Norwegian Mathematical Society), which began its meetings in 1918, and he served as the first editor of the society's journal, the Norsk matematisk tidsskrift (Norwegian Journal of Mathematics) from 1919 to 1923 and as the chairman of the Norwegian Mathematical Society from 1929 to 1934 (see JOHANSSON 1948, p. 39). He also maintained his contacts with the International Commission on Mathematical Instruction and wrote two reports on the teaching of mathematics in Scandinavia after reforms had introduced differential calculus into the curriculum in 1910 and on the training of future teachers of mathematics in Norway (HEEGAARD 1930, HEEGAARD 1933).

Heegaard held the chair as professor of geometry in Oslo until his retirement in 1941. He died on 7 February 1948 in Oslo.

Heegaard devoted a considerable part of his time and energy to popularising the sciences and to topics concerning the teaching of mathematics. Thus, he wrote popular books on astronomy and physics, travelled the vast Norwegian country, lecturing in the small communities, and lectured on the great "chiefs of the sciences" on the radio (see MUNKHOLM and MUNKHOLM 1999, pp. 942-944). These latter radio shows were produced during the German occupation of Norway and were later considered politically problematic because the access to radio receivers was restricted to families in good standing with the occupation forces and their ideology (MUNKHOLM and MUNKHOLM 1999, p. 942). He also published papers on the history of mathematics and on 'elementary' subjects in the Norwegian Journal of Mathematics. Nevertheless, Heegaard only wrote two textbooks on mathematics - one on mathematics for the naval technical education (HEEGAARD 1905), and one together with Olaf M. Thalberg on mathematical geography for the gymnasium (HEEGAARD and Thalberg 1927). In the preface to the first of these textbooks, Heegaard described his presentational style by quoting from Klein'S programme:
"Der Unterricht derjenigen, welche die Mathematik nur als ein Hülfsmittel gebrauchen wollen, soll von den anschauungsmässigen Momenten einen naiven Gebrauch machen" (HEEGAARD 1905, i-ii).
Heegaard often favoured the process of intuition over that of rigorous proof. In his lectures, he is reported to have made good use of geometrically intuitive illustrations and of using stories to make the mathematics more interesting (JOHANSSON 1948, 40).

As professor of geometry, Heegaard became involved with the publication of the collected works of Sophus Lie (1842-1899) as a co-editor together with Friedrich Engel (1861-1941). These activities occupied Heegaard to the extent that he produced few research papers of international importance after his dissertation and the chapter in the Encyklopädie der mathematischen Wissenschaften. Indeed, it seems that he saw himself more as an 'educator' in a broad sense than as a research mathematician.

Selected primary literature

M. DEHN, P. HEEGAARD 1907, Analysis situs, Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Leipzig, B.G. Teubner, vol. III.AB, chapter 3, 153-220
P. HEEGAARD 1898, Forstudier til en topologisk Teori for de algebraiske Fladers Sammenhæng, København, Det Nordiske Forlag - Ernst Bojesen
P. HEEGAARD 1905, Lærebog i Mathematik udarbejdet til Brug ved Forberedelsen til Maskinisteksamens anden Afdeling og i Maskinistskolens yngste Klasse, København, G. E. C. Gad's Universitetsboghandel
P. HEEGAARD 1909, Den internationale Matematikundervisningskommission. Foreløbig Meddelelse om Kommissionens Organisation og almindelige Arbejdsplan, Nyt Tidsskrift for Matematik, Afdeling A, Oversættelse, 37-50
P. HEEGAARD 1916, Sur l' "analysis situs", Bulletin de la société mathématique de France, 44, 161-242
P. HEEGAARD 1930, Scandinavie, l'Enseignement Mathématique, 29, 307-314
P. HEEGAARD 1933, Norvège: préparation théorique et pratique des professeurs de mathématiques de l'enseignement secondaire, l'Enseignement Mathématique, 32, 360-364
P. HEEGAARD (Ed.) 1912, Der Mathematikunterricht in Dänemark, København, Gyldendalske Boghandel - Nordisk Forlag
P. HEEGAARD, O.M. THALBERG 1927, Matematisk geografi for gymnasiet, Oslo, Gyldendal

Secondary literature

I. HECKSCHER 1912a, Fra I.M.U.K., Nyt Tidsskrift for Matematik, Afdeling A, 38-44
I. HECKSCHER 1912b, Fra I.M.U.K., Nyt Tidsskrift for Matematik, Afdeling A, 105-109
I. HECKSCHER 1913, Fra I.M.U.K., Nyt Tidsskrift for Matematik, Afdeling A, 82-85
I.M.U.K. 1914, Nyt Tidsskrift for Matematik, Afdeling A, 25, 48
I. JOHANSSON 1948, Minnetale over professor Poul Heegaard, Det norske Videnskabsakademi i Oslo: Årbok, Holdt i den mat.-naturv. klasses møte den 1. november 1948, 36-45
V.T. JØRGENSEN, J. NIELSEN 1980, Poul Heegaard (1871-1948), Dansk Biografisk Leksikon, 3rd Edn. (16 vols.), København, Gyldendahl, vol. 6, 134
E.S. MUNKHOLM, H.J. MUNKHOLM 1998, Poul Heegaard (1871-1948), dansk-norsk topolog, Normat, 46, 4, 145-169
E.S. MUNKHOLM, H.J. MUNKHOLM 1999, Poul Heegaard, in I.M. James (Ed.), History of Topology, Amsterdam, Elsevier, chapter 34, 925-946
K. RAMSKOV 1998, Fra de akademiske kulisser: universitetets matematiske stillinger 1909-17, KU Preprint series, 10
K. RAMSKOV 2004, Matematiske stillinger og karrierepleje i første tredjedel af det 20. århundrede i København, HOSTA, 16
A. ROHRBERG 1915, Der mathematische Unterricht in Dänemark, n. 2 in Beihefte zur Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht, Leipzig/Berlin, B.G. Teubner, Berichte und Mitteilungen veranlasst durch die Internationale Mathematische Unterrichtskommission, Zweite Folge
V. TRIER 1915, Fra I.M.U.K., Nyt Tidsskrift for Matematik, Afdeling A, 26, 53-77

Henrik Kragh Sørensen
Department of Science Studies
University of Aarhus - Denmark

1The following biographical facts are based on (JØRGENSEN and NIELSEN 1980) and (MUNKHOLM and MUNKHOLM 1998, MUNKHOLM and MUNKHOLM 1999) and the obituary (JOHANSSON 1948), which also contains an extensive bibliography for Heegaard.