1. The distinctive styles of Klein and Lie
Geometrical Visions
The distinctive styles
of Klein and Lie

2. Uses and Abuses of Style as an Explanatory Concept
Style: a vague and problematic notion
Particularly problematic when extended to mathematical schools, research communities or national traditions (Duhem on German vs. French science)
Or when used to discredit opponents
(Bieberbach’s use of racial stereotypes against Landau and others)

3. Types of Mathematical Creativity
Hilbert as an algebraist, even when doing geometry
Poincaré as a geometer, even when doing analysis
Weyl commenting on Hilbert’s Zahlbericht
Van der Waerden (algebraist) accounting for why Weyl (analyst) gave up Brouwer‘s intuitionism

4. Different Views of Hilbert’s Work on Foundations of Geometry
Hans Freudenthal emphasized the modern elements; how he broke the umbilical cord that connected geometry with investigations of the natural world
Leo Corry emphasizes the empiricist elements that motivated Hilbert’s axiomatic approach to geometry but also his larger program for axiomatizing all exact sciences

5. Hilbert as a Classical Geometer
Hilbert’s work can also be seen within the classical tradition of geometric problem solving (Pappus, Descartes)
Greek tradition: construction with straight edge and compass, conics (Knorr)
Descartes: more general instruments used to construct special types of algebraic curves (Bos)
Hilbert, like Descartes, saw geometric problem solving as a paradigm for epistemology

6. Methodological challenge: to develop a systematic way to determine whether a well-posed geometrical problem can be solved with specified means
Descartes showed that a problem which can be transformed into a quadratic equation can be solved by straight edge and compass
19th-century mathematicians used new methods to prove that trisecting an angle and doubling a cube could not be constructed using Euclidean tools

7. Impossibility Proofs
Ferdinand Lindemann showed in 1882 that π is a transcendental number
So even Descartes’ system of algebraic curves is insufficient for squaring the circle
Hilbert regarded this as an important result, so he gave a new proof: in 4 pages!
He emphasized the importance of impossibility proofs in his famous Paris address on “Mathematical Problems”
Not all problems are created equal: he gave general criteria for those which are fruitful

8. Hilbert’s geometric vision
Doing synthetic geometry with given constructive means corresponds to doing analytic geometry over a particular algebraic number field
Solvability of a geometric problem is equivalent to deciding whether the corresponding algebraic equation has solutions in the field
Paradigm for Hilbert’s “24th Paris problem”: to show that every well-posed mathematical problem has a definite answer (refutation of du Bois Reymond’s Ignorabimus)

9. Hilberts Schlusswort aus der Grundlagen der Geometrie

10. Die vorstehende Abhandlung ist eine kritische Untersuchung der Prinzipien der Geometrie; in dieser Untersuchung leitete uns der Grundsatz, eine jede sich darbietende Frage in der Weise zu erörtern, dass wir zugleich prüften, ob ihre Beantwortung auf einem vorgeschriebenen Wege mit gewissen eingeschränkten Hilfsmitteln möglich oder nicht möglich ist. Dieser Grundsatz scheint mir eine allgemeine und naturgemäße Vorschrift zu enthalten; in der Tat wird, wenn wir bei unseren mathematischen Betrachtungen einem Probleme begegnen oder einen Satz vermuten, unser Erkenntnistrieb erst dann befriedigt, wenn uns entweder die völlige Lösung jenes Problem und der strenge Beweis dieses Satzes gelingt oder wenn der Grund für die Unmöglichkeit des Gelingens und damit zugleich die Notwendigkeit des Misslingens von uns klar erkannt worden ist.

11. So spielt dann in der neueren Mathematik die Frage nach der Unmöglichkeit gewisser Lösungen oder Aufgaben eine hervorragende Rolle und das Bestreben, eine Frage solcher Art zu beantworten, war oftmals der Anlass zur Entdeckung neuer und fruchtbarer Forschungsgebiete. Wir erinnern nur an Abel’s Beweise für die Unmöglichkeit der Auflösung der Gleichungen fünften Grades durch Wurzelziehen, ferner an die Erkenntnis der Unbeweisbarkeit des Parallelaxioms und an Hermite’s und Lindemann’s Sätze von der Unmöglichkeit, die Zahlen e und π auf algebraischem Wege zu konstruieren.

12. Der Grundsatz, demzufolge man überall die Prinzipien der Möglichkeit der Beweise erläutern soll, hängt auch aufs Engste mit der Forderung der „Reinheit“ der Beweismethoden zusammen, die von mehreren Mathematikern der neueren Zeit mit Nachdruck erhoben worden ist. Diese Forderung ist im Grunde nichts anderes als eine subjektive Fassung des hier befolgten Grundsatzes. In der Tat sucht die vorstehende geometrische Untersuchung allgemein darüber Aufschluss zu geben, welche Axiome, Voraussetzungen oder Hilfsmittel zu Beweise einer elementar-geometrischen Wahrheit nötig sind, und es bleibt dann dem jedesmaligen Ermessen anheim gestellt, welche Beweismethode von dem gerade eingenommenen Standpunkte aus zu bevorzugen ist.

13. Klein and Lie as Creative Mathematicians
Two full-blooded geometers

14. Klein’s Universality
Felix Klein was fascinated by questions of style and discussed it often in his lectures
On a number of occasions he described Sophus Lie’s style as a geometer
Geometry, for Klein, was essentially a springboard to a way of thinking about mathematics in general
This is surely the most striking and also impressive feature in his research, which covered many parts of pure and applied mathematics

15. Felix Klein as a Young Admirer and Collaborator of Lie
Studied line geometry with Plücker in Bonn,
Protégé of Clebsch in Göttingen; projective & algebraic geometry
Met Lie in Berlin, 1869
Presented his work in Kummer’s seminar

16. Klein’s first great discovery
Lie was nowhere near as broad as Klein would become, but he was far deeper
It is only a slight exaggeration to say that Klein discovered Lie
During the early 1870s he was virtually the only one who had any understanding of Lie’s mathematics
He described how Lie spent whole days “living” in the spaces he imagined

17. On Lie’s Relationship with Klein
D. Rowe, “Der Briefwechsel Sophus Lie – Felix Klein, eine Einsicht in ihre persönlichen und wissenschaftlichen Beziehungen,” NTM, 25 (1988)1,
Sophus Lie’s Letters to Felix Klein,
, ed. D. Rowe, to appear

18. Sophus Lie: a Norwegian Hero

19. Arild Stubhaug’s Heroic Portrait of Lie, the Norwegian Patriot
Interpretation of Lie’s Life as a Triumphant Struggle
Story of Friends, Foes and Betrayal
Subsidiary Theme: French wisdom vs. German petty- mindedness

20. Sophus Lie, 1844-1899 1865-68: study at Univ. Christiania
: stipend to study in Berlin, Paris
: collaboration with Felix Klein
: Prof. in Christiania
: Leipzig
1898 return to Norway

21. On Lie’s Mathematics
Hans Freudenthal, “Marius Sophus Lie,” Dictionary of Scientific Biography.
Thomas Hawkins, “Jacobi and the Birth of Lie’s Theory of Groups,” Archive for History of Exact Sciences, 1991.

22. On Lie’s Early Work
D. Rowe, “The Early Geometrical Works of Felix Klein and Sophus Lie”
T. Hawkins, “Line Geometry, Differential Equations, and the Birth of Lie’s Theory of Groups”
In The History of Modern Mathematics, vol. 1, ed. D. Rowe and J. McCleary, 1989.

23. Lie’s Early Career
: line and sphere geometry; special contact transformations
: PDEs and line complexes; general concept of contact transformations
: Lie’s vision for a Galois theory of differential equations

24. Lie’s Subsequent Career
: first work on continuous transformation groups; classification of groups for line and plane
: return to geometry; applications of group theory to differential geometry, in particular minimal surfaces
: group-theoretic investigations and differential invariants (with Friedrich Engel beginning 1884)

25. Lie’s Subsequent Career
1886: succeeds Klein as professor of geometry in Leipzig
Continued collaboration with Engel on vol. 1 of Theorie der Transformationsgruppen
: Lie spends nine months at a sanatorium outside Hannover; leaves without having fully recovered
: works on Riemann-Helmholtz space problem

26. On the History of Lie Theory
Thomas Hawkins, Emergence of the Theory of Lie Groups. An Essay in the History of Mathematics, , Springer 2000.
Four Parts:
Sophus Lie, Wilhlem Killing,
Élie Cartan, and Hermann Weyl

27. German line geometry and French sphere geometry
4-dimensional geometries derived from 3-dimensional space

28. Julius Plücker and the Theory of Line Complexes
Plücker took lines of space as elements of a 4-dim geometry
Algebraic equation of degree n leads to an nth- order line complex
Locally, the lines through a point determine a cone of the nth degree
Counterpart to French sphere geometry

29. Lie and Klein: geometries based on free choice of the space elements
Line and sphere geometry were central examples
Klein also studied spaces of line complexes in 1860s, the space of cubic surfaces (1873), etc.
In his Erlangen Program he emphasizes that the dimension of the geometry is insignificant, since one can always let the same group act on different spaces obtained by varying the space element, which may depend on an arbitrary number of coordinates

30. Kummer surfaces and their physical and geometrical contexts

31. Kummer Surfaces
Quartic surfaces with 16 double points (here all are real)
Klein was the first to study these as the singularity surfaces that naturally arise for families of 2nd-degree line complexes

32. The Fresnel Wave Surface
Kummer‘s study of ray systems revealed that the Fresnel surface was a special type of Kummer surface
It has 4 real and 12 complex double points

33. Lie’s Breakthrough, Summer 1870
Line-to-sphere transformation
Maps the principle tangent curves of one surface onto the lines of curvature of a second surface
Lie applied this to show that the principle tangent curves of the Kummer surface were algebraic curves of degree 16
Klein recognized that they were identical to curves he had obtained in his work on line geometry

34. Klein’s Correspondence with Lie
Used by Friedrich Engel in Band 7 of Lie’s Collected Works
Fell into Hands of Ernst Hölder, son of Otto Hölder, who married one of Lie’s granddaughters
Purchased by the Oslo University Library
To be published by Springer in a German/English edition

35. Klein’s letters to Lie, 1870-1872
Collaboration in Berlin and Paris,
Klein had trouble following Lie’s ideas by 1871
Lie’s visit in summer led to enriched version of Klein’s Erlanger Programm

36. Klein’s Style as a Geometer

37. Felix Klein as a Young Admirer of Riemann
Came in Contact with Riemann’s Ideas through Clebsch in Göttingen ( )
Competed as self- appointed champion of Riemann with leading members of the Weierstrass school

38. Alfred Clebsch ( )
Leading „Southern German“ mathematician of the era
Founder of Mathematische Annalen
Klein was youngest member of the Clebsch School

39. Klein’s “Physical Mathematics”
Accounting for the Connection between singular points and the genus of a Riemann surface

40. Klein (borrowing from Maxwell) to Visualize Harmonic Functions

41. Building complex functions on an abstract Riemann surface
Rather than introducing complex functions in the plane and then building Riemann surfaces over C, Klein began with a non- embedded surface of appropriate genus
The harmonic functions were then introduced using current flows as before
He visualized their behavior under deformations that affected the genus of the surface

43. Klein on Visualizing Projective Riemann Surfaces
Mathematische Annalen,

44. Identifying Real and Imaginary Points on Real Algebraic Curves
Riemann and Clebsch had dealt with the genus of a curve as a fundamental birational invariant
Klein wanted to find a satisfying topological interpretation of the genus which preserved the real points of the curve
He did this by building a projective surface in 3-space around an image of the real part of the curve in a plane

45. Carl Rodenberg‘s Models for Cubic Surfaces

46. The Clebsch Model for a „Diagonal Surface“
Klein studied cubics with Clebsch in Göttingen in 1872
Clebsch came up with this special case of a non- singular cubic where all 27 lines are real
There are 10 Eckhard points where 3 of the 27 lines meet

47. Klein on Constructing Models (1893)
„It may here be mentioned as a general rule, that in selecting a particular case for constructing a model the first prerequisite is regularity. By selecting a symmetrical form for the model, not only is the execution simplified, but what is of more importance, the model will be of such a character as to impress itself readily on the mind.“

48. Klein on his Research on Cubics
„Instigated by this investigation of Clebsch, I turned to the general problem of determining all possible forms of cubic surfaces. I established the fact that by the principle of continuity all forms of real surfaces of the third order can be derived from the particular surface having four real conical points ”

49. A Cubic with 4 singular points
Klein began by considering a cubic with 4 singular points located in the vertices of a tetrahedron
The 27 lines collapse into the 6 edges of the tetrahedron

50. Removing Singularities by Deformations
Two basic types of deformations
The first splits the surfaces at the singular points
The second enlarges the surface around the singularity

51. Moving about in the Space of Cubic Surfaces
The nonsingular cubics form a 19- dimensional manifold
Those with a single conical point form an 18-dimensional submanifold, and so on
So starting with the special point in the 15- dimensional submanifold with 4 singularities, Klein could move up step by step through the entire manifold to exhaust the classification

52. Vision behind this research
“What is of primary importance is the completeness of enumeration resulting from my point of view; it would be of comparatively little value to derive any number of special forms if it cannot be proved that the method used exhausts the subject. Models of the typical cases of all the principal forms of cubic surfaces have since been constructed by Rodenberg for Brill’s collection.”

53. Some Stylistic Elements in Lie’s Early Work

54. Scheffers’ editions of Lie’s lectures
: Georg Scheffers writes three books based on Lie’s lectures:
1) DEQs with known infinitesimal Transformations (1891)
2) Continuous Groups (1893)
3) Geometry of Contact Transformations (1896)

55. Solving Differential Equations
According to Engel, Lie had already realized in 1869 that an ordinary first-order DEQ
can be reduced to quadratures if one can find a one-parameter family of transformations that leaves the DEQ invariant.

56. By 1872 Lie saw that it was enough to have an infinitesimal transformation that generated the 1-parameter group. Thus if the DEQ
admits a known infinitesimal transformation
in which, however, the individual integral curves do not remain invariant, then the DEQ has an integrating factor.

57. The integrating factor
then leads directly to a solution by quadrature in the form:

58. Lie’s geometric interpretation of the integrating factor

59. Lie’s Work on Tetrahedral Complexes
A tetrahedral line complex consists of the lines in space that meet the four planes of a coordinate tetrahedron in a fixed cross ratio
Such complexes were studied earlier by Theodor Reye and so were sometimes known as “Reyesche Komplexe”
Lie generated such complexes by letting a 3-parameter group act on a given line

60. Lie and Klein study W-Kurven
Earliest jointly published work of Lie and Klein dealt with W-Kurven (W = Wurf, an allusion to Staudt’s theory)
Such curves in the plane are left invariant by a 1-parameter subgroup of the projective group acting on the plane
They work on W-Kurven and W-Flächen in space, but find this too complicated and tedious, so they never finish their manuscript

61. Lie’s interest in geometrical analysis
Lie studied surfaces tangential to the infinitesimal cones determined by a tetrahedral complex, which leads to a first- order PDE of the form:

62. Lie used a special transformation to map this DEQ to a new one
which was left invariant by the 3- parameter group of translations in the space (X,Y,Z). This enabled him to reduce the equation to one of the form
which could be solved directly.

63. This result soon led Lie to the following insights:
1) PDEs that admit a commutative 3-parameter group can be reduced to the form
2) PDEs that admit a commutative 2- parameter group can be reduced to
3) PDEs that admit a 1-parameter group can be reduced to

64. Lie’s Theory of Contact Transformations
Lie noticed that the transformations needed to carry out the above reductions were in all cases contact transformations.
Earlier he had studied these intensively, in particular in connection with his line-to- sphere transformation.

65. Lie’s Surface Elements
For a point (x,y,z) on a surface F given by z = f(x,y), the equation for the tangent plane is
For an infinitely small region, Lie associated to each point (x,y,z) of F the surface element with coordinates (x,y,z,p,q). All 5 coordinates are treated equally.

66. The following local condition holds:
and describes the property that contiguous surface elements intersect. This Pfaffian relation must hold under an arbitrary contact transformation.
Lie had no trouble extending these notions to n-dimensional space in order to deal with PDEs of the form:

67. Lie then (1872) defined a general contact transformation analytically as a mapping
for which the condition
remains invariant. He showed further that two first-order PDEs can be transformed to another by means of a contact transformation.

68. Lie’s Adaptation of Jacobi’s Theory
In his “Nova methodus” Jacobi introduced the bracket operator
within his theory of PDEs. This was a crucial tool for reducing a non-linear PDE to solving a system of linear PDEs.

69. Lie’s Notion of PDEs in Involution
Lie interpreted the bracket operator geometrically, borrowing from Klein’s notion of line complexes that lie in involution. He defined two functions
to be in involution if

70. Lie’s First Results on Differential Invariants
Lie showed that a system of m PDEs
satisfying
remains in involution after the application of a contact transformation.
Such considerations led Lie to investigate the invariant theory of the group of all contact transformations.

71. On the Reception of Lie’s Work

72. Berlin Reactions to Lie’s Work
Weierstrass considered Lie’s work so wobbly that it would have to be redone from the ground up
Frobenius claimed Lie’s approach to differential equations represented a retrograde step compared with the elegant techniques Euler and Lagrange

73. Freudenthal on Lie’s failure to find an adequate language
Lie tried “to adapt and express in a host of formulas, ideas which would have been better without them. . . . [For] by yielding to this urge, he rendered his theories obscure to the geometricians and failed to convince the analysts.”
The three volumes written by Engel had a distinctly “function-theoretic touch”

74. Where to look for Lie’s Vision
According to his student Gerhard Kowalewski, Lie never referred to the volumes ghost-written by Engel but rather always cited his own papers
This suggests that the “true Lie”—to take up Klein’s image—should not be sought in the volumes produced with Engel’s assistance but rather in his own earlier papers and his lectures as edited by Georg Scheffers

75. Lie’s Break with Klein

76. Lie’s Preface from 1893 Thanks those who helped pave his way:
Course with Sylow on Galois theory (1863)
Clebsch, Cremona, Klein, Adolf Mayer, and “especially Camille Jordan”
Darboux for promoting his geometrical work
Picard, first to recognize importance of Lie’s group theory for analysis
J. Tannery for sending students from ENS
Engel and Scheffers for writing his books

77. Lie on Poincaré’s Support
Lie expressed his gratitude to Poincaré for his interest in numerous applications of group theory. He was “especially grateful that he [Poincaré] and later Picard stood with me in my fight over the foundations of geometry, whereas my opponents tried to ignore my works on this topic.” (In the text one learns who these “opponents” were.)

78. Klein’s Erlangen Program
A supplement to Tom Hawkins, “The Erlanger Programm of Felix Klein: Reflections on its Place in the History of Mathematics,” Historia Mathematica 11 (1984):

79. Klein’s Lectures on Higher Geometry
Circa 1890 Klein was returning to several topics in geometry he had pursued twenty years earlier in collaboration with Lie
Corrado Segre had Gino Fano prepare an Italian translation of the Erlangen Program
Soon afterward it appeared in French and English translations
Klein wanted to republish it in German too, along with several of Lie’s earlier works

80. End of a Partnership
Klein even wrote two drafts for an introductory essay on their collaboration during the period 1869–1872
Lie profoundly disagreed with Klein’s portrayal of these events
He also realized that his own subsequent research program had little to do with the Erlangen Program
Lie felt under appreciated in Germany and from 1889–1892 was severely depressed

81. Lie on Klein and the “Erlangen Program” from 1872
Words that Scandalized the
German Mathematical Community

82. “Felix Klein, to whom I communicated all of my ideas in the course of these years [ ], developed a similar point of view for discontinuous groups. In his Erlangen Program, where he reported on his and my ideas, he speaks beyond this of groups that are neither continuous nor discontinuous in my terminology, for example he speaks of the group of Cremona transformations That there is an essential difference between these types of groups and those I have named continuous groups, namely that my continuous groups can be defined by differential equations, whereas this is not the case for the former groups, evidently escaped him completely.”

83. “Moreover, one finds hardly a trace of the all important concept of differential invariant in Klein’s Program. Klein took no part in creating these concepts, which first make it possible to found a general theory of invariants, and it was only from me that he learned that every group defined by differential equations determines differential invariants that can be found by integration of complete systems.”

84. Lie felt compelled to clarify these matters because “Klein’s pupils and friends have continually represented the relationship between Klein’s works and mine falsely,” and also because some of Klein’s remarks appended to the recently reissued Erlangen Program could easily be misconstrued.
“I am not a pupil of Klein, nor is the reverse the case, even though it perhaps comes closer to the truth I rate Klein’s talent highly and will never forget the sympathy with which he followed my scientific efforts from the beginning, but I believe that he, for example, does not sufficiently distinguish between induction and proof, between the introduction of a concept and its utilization.”

85. Seeking New Allies
These remarks scandalized many within Klein’s extensive network (Hilbert, Minkowski)
But Lie also criticized several others by name, including Helmholtz, de Tilly, Lindemann, and Killing
He also singled out several French mathematicians for praise