The Mathematician Sophus Lie: It was the Audacity of my Thinking

Description

Sophus Lie (1842-1899) is one of Norways greatest scientific talents. His mathematical works have made him famous around the world no less than Niels Henrik Abel. The terms "Lie groups" and "Lie algebra" are part of the standard mathematical vocabulary. In his comprehensive biography the author Arild Stubhaug introduces us to both the person Sophus Lie and his time. We follow him through: childhood at the vicarage in Nordfjordeid; his youthful years in Moss; education in Christiania; travels in Europe; and learn about his contacts with the leading mathematicians of his time.

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Review

"... Es ist besonders erfreulich, dass sich der Autor zum Ziel gesetzt hat, nach Abel auch Lie einem über die Mathematiker weit hinausgehenden Leserkreis nahe zu bringen. In seinem lesenswerten Buch stellt er den interessanten Lebenslauf eines nicht immer bequemen Genies voll irrtümlicher Lebenskraft, abe auch mit Schattenseiten dar. ... Ein spannendes Buch, das auch Einblick in die deutsche Universitätsgeschichte des späten 19. Jahrhunderts gibt, aber auch ein tröstliches und ermutignedes Buch für junge Mathematiker. ... Zusammenfassend, ein Buch, das man gerne im eigenen Bücherregal stehen hat." (P. Gruber, Internationale Mathematische Nachrichten 2003, Heft 57, Ausgabe 192)

"... Ich muss allerdings eingestehen, dass sich mir, als ich das sorgfältig und mit Liebe aufgemachte Buch zum ersten Mal in Händen hielt, die Frage aufdrängte, wen denn dieses Buch interessieren könnte. Beim ersten Lesen ertappte ich mich oft dabei, dass ich Passagen, die nicht unmittelbar mit seiner Tätigkeit als Mathematiker zu tun hatten, diagonal las oder zu überspringen suchte, in der Hoffnung eingeweiht zu werden in eine Gedankenwelt, der die bahnbrechenden neuen, partielle Differentialgleichungen und Geometrie vereinenden Ideen entsprangen, die zur Theorie der Transformationsgruppen führten. Das fand ich zwar nicht: wie in der Biographie über N.H. Abel vom gleichen Autor(siehe die Besprechung von E. Behrends) ist auch dieses Buch von Stubhaug ohne eine einzige Formel oder präzise mathematische Definition. Was aber Stubhaug uns durch seine umfangreichen Nachforschungen, hauptsächlich in Briefen Lies an seine Frau, an seine Freunde in Norwegen und Briefe an und von seinen Kollegen, mitteilen kann, lässt in vielen Einzelheiten, Stück für Stück, ein breiteres Bild von Lie, der wissenschaftlichen und politischen Situation Norwegens zu Lies Zeit, und des wissenschaftlichen und gesellschaftlichen Lebens an Lies Wirkungsstätten entstehen. Die Darstellung, in der zum Teil die Inhalte einer Reihe von Briefen nacheinander auch mit Hilfe kurzer Zitate referiert werden, birgt die Gefahr der Ermüdung. Dem entgeht Stubhaug einmal durch seinen angenehmen Erzählstil, zum anderen durch gelegentliche Unterbrechungen, in denen der zeitliche und geschichtliche Hintergrund angesprochen wird. Jedenfalls fiel mir auf, dass ich mit wachsender Neugierde die zu Beginn überflogenen Stellen eine nach der anderen wieder aufsuchte und mit Interesse las. ..........

Das Buch bringt allen großen Gewinn, die sich für Sophus Lie und seine Zeit interessieren. Insbesondere erfährt man viel über den wissenschaftlichen Kontakt zwischen Lie und anderen mathematischen Größen der damaligen Zeit wie Klein, Kummer, Study, Darboux, Poincar und anderen. Wen die mathematischen Ideen Lies interessieren, der kommt nicht umhin, sich mit dessen Arbeiten auseinanderzusetzen, um dann in Verbindung mit Stubhaugs Buch vielleicht doch etwas von der "audacity of my (Lie's) thinking" zu erblicken."

P.S. Man sollte vielleicht auch erwähnen, dass das Buch im Vergleich zu Mathematikbüchern, die ähnlich aufwendig gestaltet sind, außergewöhnlich preiswert ist." (E. Vogt, http://www.mathematik.de)

The study of the history of Lie groups has been extraordinarily fortunate in recent years, being blessed first with T. Hawkins' definitive history of the mathematical theory [Emergence of the theory of Lie groups, Springer, New York, 2000; MR1771134], and now with this comprehensive and definitive life of the great Norwegian genius, a translation from Norwegian of a biography published in 2000. Where Hawkins' essay covers some 60 years of mathematical developments and is very explicit about the mathematics, the present essay is aimed at the reader who wants pure biography. There are no mathematical formulas in the text, and Lie's work is described in general terms, as it affected his life. However, this book contains a lengthy discussion of Lie's results and what they meant to the mathematical world of the time. In fact, the great charm of this work for the reviewer lies in the way it brings alive both the world of Norwegian society in the mid-nineteeenth-century and the vibrant mathematical life in Germany, France, and Italy in the late nineteenth century.
The book is divided into seven parts, with a number of appendices. Part I gives a summary of Lie's life, such as one might find in an encyclopedia. This sketch helps to orient the reader for the extremely rich and detailed biography that follows in the other six parts. Part II gives the history of Lie's ancestors on both his mother's and his father's side in the two centuries that preceded his birth, showing how the family, which had originally supported itself by fishing and other labor, became a part of the intellectual/clerical class. Lie's father, the vicar of Nordfjordeid, seems to have been an altogether admirable and forceful man, a veritable Monsieur Madeleine, under whose leadership the entire village became more sober and industrious. The move from Nordfjordeid to Moss, when Lie was nine years old, was the boy's first great travel adventure. However, within a few years of this move, his mother and younger brother died, and his father's powers suffered a decline. Part III gives the details of Lie's early education, at a time when Latin and Greek were being displaced by science in the curriculum. Lie was an excellent student in everything but Greek, at which he was merely "very good''. He grew into a physically vigorous young man, with a great love of the out-of-doors and the usual interests of a young man, about which he wrote to his father with amazing candor. Part IV discusses Lie's introduction to modern geometry. Here the author mentions the societal resistance to geometries that seemed to have no relation to physical space, a resistance frequently based on religion, and paralleling the resistance to the theory of evolution. (This resistance continued well into the twentieth century, when a priest, the Rev. J. J. Callahan, President of Duquesne University, claimed to have refuted all non-Euclidean geometry [Euclid or Einstein, Devin-Adair, New York, 1931].) This part tells of Lie's first trip to the Continent in 1870 and ends with his engagement to Anna Birch (proposed and accepted by letter) in 1872. Part V covers the years of his professorship in Christiania (Oslo) from 1872 through 1888, neatly fitting his mathematical work into the context of both his own life and the European mathematical world of the time. Part VI discusses the Leipzig years, from 1888 to 1898. These, of course, were the most event-filled years of Lie's life, both professionally and personally, in which he suffered a nervous breakdown, which was treated by the doctors with opium, leading to an addiction that Lie overcame by sheer force of will. These were also the years when Lie produced his three-volume masterpiece on transformation. His break with Klein over what he felt was Klein's refusal to give him adequate credit for his part in the Erlangen Programme is discussed in detail. (Probably this painful break accounts for the absence of a biographical sketch of Lie in Klein's [Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, Springer, Berlin, 1979; MR0529278]. Klein of course discusses Lie's work, but he does so piecemeal, in a number of different places; and in discussing the Erlangen Programme he merely refers to what "Lie and I'' contributed to the formulation of the notion of an abstract group. Part VII covers the final year of Lie's life, up to his death from pernicious anemia (the disease from which Hilbert was later to be rescued by the timely discovery of a cure). Four excellent appendices give a chronology of important events in Lie's life, a full set of notes and comments on each chapter of the text, a complete list of Lie's publications, and the story of his children and grandchildren. An outstanding feature of the book is the exceedingly rich collection of period photographs, of places in Norway significant to Lie's life, of Lie's family, of famous mathematicians of the time, reproductions of important letters and newspaper clippings, and sketches of various people (including several of Lie himself, one made the day after his death). There are eight beautiful color reproductions, seven of paintings from the period, and one of Leipzig University.
The book has been translated into very good English, for which the reviewer (who would be able to read the original only with difficulty, a dictionary in hand) is very grateful. The translator and copy editor are to be praised for their work. Only in a few places is the connotation slightly wrong. The reviewer at first misread the author's intention on p. 308, believing that Lie had recommended the rejection of an article by Klein and that Mittag-Leffler had published it anyway. Actually Mittag-Leffler had rejected the article without Lie's knowledge. The English is not wrong here, but the words "should have'' are ambiguous. Interestingly, the title and subtitle (a quotation from an 1893 letter of Lie) of the Norwegian original seem to have been interchanged in the translation, and this change was obviously intentional. Given the recent success of the book and film A beautiful mind, comparisons with John Nash come immediately to mind, and there are certainly a number of interesting parallels in both the personal and professional lives of Lie and Nash. However, Lie is certain to be incomparably the more influential of the two. It is a pity that this admirable biography will probably not be seized upon by the film industry. Perhaps, however, one might dare hope for a documentary from the BBC in Great Britain or PBS in the United States.

Reviewed by R. L. Cooke