EL COCINERO, EL LADRÓN, SU MUJER Y SU AMANTE:

In the period when Camille Jordan was absorbed by his book and inspired by relevant ideas, two young mathematicians were studying with him. They had finished their university studies and had come to Paris to enlarge their vision and begin independent research. They were the Norwegian Sophus Lie and the German Felix Klein. Their position was that of Jordan's postgraduate students, and they proved to be fine pupils indeed. As fate would have it, Lie's and Klein's studies with Jordan lasted for a very short time. However, they struck deep roots, and the ideas of Galois and Jordan played a crucial role in the subsequent scientific careers of both mathematicians.

Sophus Lie was born in 1 842 into the family of a pastor in Norway. His childhood was passed in his parents' home on the shore of the ocean near Bergen. He travelled the length and breadth of the country on foot and all his life retained a passionate love for the beauty of Norwegian fiords and Norway's natural scenes. At school Lie mastered all subjects equally well, and after finishing school was at first unable to choose an occupation. His father wanted him to follow in his footsteps and become a pastor, and Sophus gave serious thought to studying theology. It was much later, after considerable thought and not without painful doubts, that he undertook the study of mathematics and natural sciences. At first, his studies at Christiania University failed to put an end to his doubts. The breakthrough came in 1 868, when Lie read the works of V. Poncelet and J. Plucker (to which we will return below).

These outstanding geometers made the strongest impression on the young Lie. Their works led to his first publications, which were followed by a continuous stream of papers, uninterrupted for several decades. To continue his education Lie moved to Berlin in 1 870. There he met and immediately made friends with Klein, who was seven years his junior; the first joint work by Lie and Klein, described below, comes from the same year. The close personal and scientific relationship between Lie and Klein, which began then in Berlin, played a major role in the life of both mathematicians and continued until Lie's death.

The two made a visit to Paris, prompted by their desire to meet Jordan and also Gaston Darboux (1824- 1917). Darboux was the best known specialist in

Gaston Darboux

differential geometry, which applies the differential calculus to the study of local properties (i.e., properties dealing only with small neighborhoods of a point) of curves and surfaces. 50 Darboux's voluminous and profound works (mention should be made, above all, of Lefons sur Ia theorie generale des su�faces et les applications geometriques du calcul infinitesimal, Vol. 1 -4, Paris, Gauthier-Villars, 1 887- 1 896; second edition 1914-1925) influenced both Klein and, especially, Lie. In particular, many of Lie's works were inspired by the approach of the General Theory of Surfaces, which organically combines differential geometry and the theory of differential equations. Here geometric questions are very efficiently reduced to analytic ones, and both approaches are used to study differential equations. All of this compels us to write about Darboux in greater detail.

Darboux was born in Nimes in the south of France, but his whole life as a researcher and teacher was associated with Paris. He lived continuously in Paris from the age of eighteen and played an outstanding role in its intellectual life, above all as head of L'lnstitut and, as such, a member of the French Academie (see Notes 65 and 1 76). Darboux's name is linked to a considerable degree with the flourishing of the Ecole Normale, as well as the tradition whereby all outstanding French mathematicians taught in secondary school after graduating from college. Darboux was virtually the first outstanding mathematician to study at the Ecole Normale (a teachers' college), then a school less well known than the Ecole Polytechnique (see Chapter 1). Sub­

sequently he taught at the Ecole Normale for many years. The respect enjoyed by Darboux even in government circles soon proved to be of great use to Lie

24 Felix Klein and Sophus Lie

(see below). In other cases, however, Darboux's influence was less favorable:

for example Darboux, somewhat conservative in his mathematical tastes, opposed the defense of Henri Leon Lebesgue's (1875-1941) doctoral thesis.

Only the influence of Emile Picard (1856-1941), Darboux's future successor as presideht of the Institut, sanctioned the defense of Lebesgue's thesis, which was to play an outstanding role in twentieth-century mathematics.

Klein and Lie were not destined to remain in Paris for a long time; neverthe­

less, the personal contacts of both mathematicians with Jordan (and Darboux) played an enormous role in their subsequent research (see Chapter 8). Actually, the two friends were planning to stay in Paris long enough to become familiar with the main achievements of the French mathematical school, and then move on to London for contacts with English mathematicians. The Franco­

Prussian war broke out in 1871 and the German Klein had to leave France in a hurry. (0 idyllic age when Klein was not even detained in Paris and could freely leave for Germany!) He intended to reenter France with the Prussian troops, but his military career was aborted-he contracted typhus and, in the meantime, France was rapidly routed. Left without his friend, Lie-who was an experienced hiker-decided to take advantage of the forced interruption in their studies to make a trek through all of France, the Alps, and Italy. But in the wartime atmosphere the plan proved to be rather unfortunate. Because of his poor French, conspicuous height and handsome but purely Nordic appearance, 5 1 Lie was immediately arrested as a German spy and imprisoned.

Apparently, French patriots found Lie's manner of looking around in an abstracted way (he was then thinking through some mathematical problem) and then fevereshly scribbling in a little notebook (he was making mathemati­

caljottings-in Norwegian) extremely suspicious. He spent about a month in the prison of Fontainebleau (just southwest of Paris-pretty far from the Alps!). As soon as he learned of Lie's arrest, Darboux used all his contacts in order to have Lie freed. But conditions in prison were not particularly bad, and Lie spent the time pondering over some aspects of Plucker's line geometry, to which his attention had been drawn by Klein and about which we shall say more below. Upon being freed from prison, Lie, the tireless hiker, went on his trek through France and Italy.

While Lie could be described as a typical nineteenth-century scholar, his friend and colleague Felix Klein was a very different individual, both in his attitude to science and in character. A born leader, a brilliant polemicist, a great teacher, and an excellent organizer, capable of implementing the most complex schemes and undertakings, Klein was a precursor of twentieth­

century science. Klein combined the qualities of organizer, teacher, and researcher to a remarkable extent. (Some of his modern counterparts are, say, the Parisian Jean Alexandre Dieudonne (born in 1906), one of the leaders of the Bourbaki group, and the Moscow physicist Lev Landau (1908-1968).)

In the relationship between Lie and Klein, the latter, who was the younger of the two, played the role of the elder. Thus it was on Klein's initiative that the friends set out from Berlin to Paris and London (at that time they failed

to reach London), it was Klein who (years later) suggested that Lie move from Norway to Germany (Leipzig), and so on. Klein's leadership was readily accepted by the unassuming and kindhearted Lie although, perhaps, deep down he felt slighted. Like any truly outstanding scientist, Lie was well aware of the value of his work and was proud of it (his works will be dealt with in detail in Chapter 6). Lie also knew that from the purely scientific standpoint his influence on Klein was greater than Klein's on him. In any case, it was a question of scientific primacy which provoked the only conflict between Lie and Klein in their otherwise remarkably smooth friendship. 52

Klein was born in 1849 in Dusseldorf into the family of an official in the finance department. His father held extremely conservative, old-Prussian views; Felix adhered to some while flatly rejecting the others. In accordance with his father's wishes, Klein studied at a classical gymnasium. There, much attention was paid to ancient languages and very little to mathematics and the natural sciences. The deep antipathy Klein developed for the gymnasium 53 played an important role in his future pedagogical views. After graduating from the gymnasium, Klein entered the university in Bonn. There he was immediately noticed and singled out by Julius Plucker (1801-1868), who headed the departments of (experimental) physics and (pure) mathematics. In 1866, the seventeen-year-old Klein became Plucker's assistant in the physics department.

Plucker intended to make a physicist out of Klein. The latter showed a lively interest in physics (he was very "physics-minded" -more about that below) and had no objections to Plucker's intentions. But these plans were not destined to be carried through. In 1868 Plucker died, and it fell to Klein's lot to carry out the painstaking job of preparing for publication his mentor's unfinished works, above all the second part of the remarkable Neue Geometrie des Raumes, gegrundet auf die Betrachtung der geraden Linien als Raum­

element. Work on the book (published in 1869) inspired Klein, and his first series of independent papers grew out of it, contributing to Klein's develop­

ment as a mathematician.

After Plucker's death, Klein lost his post as an assistant, left Bonn and went to Gottingen and to Berlin where he became acquainted with the young but very influential Gottingen mathematician Rudolf Friedrich Alfred Clebsch (1833-1872), the physicist Wilhelm Weber (1804-1891), a friend and colleague of the great Gauss, 54 and the head of the Berlin school of mathematics Karl Theodor Wilhelm Weierstrass (1815-1897). It should be noted that whereas Klein's relations with Clebsch and Weber were quite friendly from the begin­

ning, his relationship with Weierstrass was marked from the outset by barely concealed antipathy on both sides. The roots of that hostility lay in the total incompatibility of Klein's and Weierstrass's scientific positions. This deserves a more detailed explanation.

It is now well known that the human brain is not symmetric, and that the left and right hemispheres of the cerebrum each have their specific functions.

The great interest focussed at present on a range of a questions having to do

26 Felix Klein and Sophus Lie

with that asymmetry was reflected, in particular, in the award of the 1981 Nobel Prize in biology and medicine to the American psychologist Roger Sperry for research in that field. In the standard case of the right-handed person, the left hemisphere is responsible for analytic, logical thinking, while the right hemisphere provides the "pictorial," synthetic vision of the world (of course, here we describe the differences between the hemispheres roughly and incompletely). With the same degree of approximation as above, it can be said that the left hemisphere, undoubtedly linked with speech, writing, 55 as well as computation and the use of the set of natural numbers in general, controls the algebraic aspects of mathematics (since algorithmic procedures have to do with linearly arranged algebraic formulas, they undoubtedly belong to the domain of the left hemisphere), while the right hemisphere is associated with geometric vision, with diagrams and pictures. Possibly, however, it may be more correct to link the left hemisphere with logic, and the right one with physics, bearing in mind the global approach to nature and to the phenomena of the natural sciences characteristic of most physicists.

The above may go part of the way in explaining the striking fact of the existence of two types of mathematicians, opposite in some respects: alge­

braists, whose thinking has to do primarily with logic, formulas, and algo­

rithmic procedures; and geometers or physicists, who proceed mostly from graphic and visual impressions rather than from formulas. The existence of these two types of poorly correlated approaches to mathematics, as well as the existence of scientists for whom one or the other dimension in mathematics is the dominant one, was pointed out in a lecture56 by the eminent mathe­

matician Hermann Weyl (1885-1955) (we shall come across his name again below). 57 In that lecture Weyl mentioned Klein and B. Riemann (see below) as examples of "physicists" and Weierstrass as an "algebraist." An earlier example is presented by the founder of the differential and integral calculus, the great physicist Isaac Newton (1642-1727) on the one hand, and the great logician Gottfried Wilhelm Leibniz (1646-1716) on the other. It is plausible that the mutual antipathy developed by Klein and Weierstrass was fostered by analogous differences in their respective scientific outlooks. 58

Klein's "physical" thinking has already been mentioned; it was reflected in many of his research papers, for example, in the remarkable Lectures on Riemann Surfaces (a course of lectures delivered in Gottingen and circulated in mimeographed form) in which Klein took the liberty of considering the distribution of electric charges along a conductor shaped as an abstract Riemann surface of extremely complex topological structure in order to prove purely mathematical theorems. Klein's teaching (see Note 60) was also characterized by a physical and graphic approach, and, in consequence, by a certain lack of rigor. Klein's mode of exposition was largely due to the in­

fluence of the great Bernhard Riemann. Klein worshipped Riemann, whereas Weierstrass, the fanatic advocate of rigor (modern mathematics largely owes its spirit and style to him), assailed Riemann and his friend Lejeune Dirichlet and considered many of their results unproved or even incorrect. This

con-structive criticism gave rise to the theory of real numbers and to many topological concepts. In this connection, an interesting incident was recounted (at second hand) by Arnold Sommerfeld (1868-1951), one of the greatest of twentieth-century physicists. Sommerfeld was Klein's pupil and staff member for many years; Klein engaged him as his assistant, just as he himself had once been appointed assistant in the physics department by Plucker. 59 Sommer­

feld told how, in the early 1860s, Weierstrass and the outstanding German physicist, mathematician, biologist, and medical doctor Hermann Helmholtz (1821-1894) spent a summer together in the country. Weierstrass had taken along Riemann's famous work (the source of the entire modern theory of functions of a complex variable) in order to elaborate and analyze it in his free time-while the highly "physicsminded" Helmholz could not imagine what there was to elaborate (see A. Sommerfeld, "Klein, Riemann und die mathe­

matische Physik" Naturwissenschaft, 7 (1919), 300-303).60

Klein's close friendship with Lie and the latter's significant scientific in­

fluence on him made up for the absence of fruitful scientific contacts with Weierstrass. We have already described Lie and Klein's joint trip to Paris, which played an important role in the careers of both mathematicians. After returning from France and recovering from typhus, Klein settled in Gottingen not far from Clebsch and Weber; this was an extremely productive time for him. However, before dealing in detail with the scientific achievements of Klein and Lie, it is necessary to describe briefly the scientists who laid the foundation for their successes.

CHAPTER 3