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Geometrical Visions The distinctive styles of Klein and Lie

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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 (Bieberbachs use of racial stereotypes against Landau and others)

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

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Different Views of Hilberts 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 Hilberts axiomatic approach to geometry but also his larger program for axiomatizing all exact sciences

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Hilbert as a Classical Geometer Hilberts 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

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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

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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

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Hilberts 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 Hilberts 24th Paris problem: to show that every well-posed mathematical problem has a definite answer (refutation of du Bois Reymonds Ignorabimus)

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Hilberts Schlusswort aus der Grundlagen der Geometrie

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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.

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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 Abels Beweise für die Unmöglichkeit der Auflösung der Gleichungen fünften Grades durch Wurzelziehen, ferner an die Erkenntnis der Unbeweisbarkeit des Parallelaxioms und an Hermites und Lindemanns Sätze von der Unmöglichkeit, die Zahlen e und π auf algebraischem Wege zu konstruieren.

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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.

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Klein and Lie as Creative Mathematicians Two full-blooded geometers

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Kleins Universality Felix Klein was fascinated by questions of style and discussed it often in his lectures On a number of occasions he described Sophus Lies 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

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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 Kummers seminar

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Kleins 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 Lies mathematics He described how Lie spent whole days living in the spaces he imagined

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

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Sophus Lie: a Norwegian Hero

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

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Sophus Lie, : study at Univ. Christiania : stipend to study in Berlin, Paris : collaboration with Felix Klein : Prof. in Christiania : Leipzig 1898 return to Norway

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

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On Lies Early Work D. Rowe, The Early Geometrical Works of Felix Klein and Sophus Lie T. Hawkins, Line Geometry, Differential Equations, and the Birth of Lies Theory of Groups In The History of Modern Mathematics, vol. 1, ed. D. Rowe and J. McCleary, 1989.

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Lies Early Career : line and sphere geometry; special contact transformations : PDEs and line complexes; general concept of contact transformations : Lies vision for a Galois theory of differential equations

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Lies 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)

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Lies 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

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

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

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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

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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

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Kummer surfaces and their physical and geometrical contexts

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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

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The Fresnel Wave Surface Kummers 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

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Lies 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

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

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Kleins letters to Lie, Collaboration in Berlin and Paris, Klein had trouble following Lies ideas by 1871 Lies visit in summer 1872 led to enriched version of Kleins Erlanger Programm

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Kleins Style as a Geometer

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

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

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Kleins Physical Mathematics Accounting for the Connection between singular points and the genus of a Riemann surface

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Klein (borrowing from Maxwell) to Visualize Harmonic Functions

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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

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Klein on Visualizing Projective Riemann Surfaces Mathematische Annalen,

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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

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Carl Rodenbergs Models for Cubic Surfaces

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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

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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.

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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....

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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

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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

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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

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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 Brills collection.

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Some Stylistic Elements in Lies Early Work

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

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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.

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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.

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

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Lies geometric interpretation of the integrating factor

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Lies 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

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Lie and Klein study W-Kurven Earliest jointly published work of Lie and Klein dealt with W-Kurven (W = Wurf, an allusion to Staudts 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

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Lies 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:

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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.

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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

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Lies 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.

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Lies 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.

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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:

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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.

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Lies Adaptation of Jacobis 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.

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

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Lies 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.

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On the Reception of Lies Work

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

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Freudenthal on Lies 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

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Where to look for Lies 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 Lieto take up Kleins imageshould not be sought in the volumes produced with Engels assistance but rather in his own earlier papers and his lectures as edited by Georg Scheffers

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Lies Break with Klein

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Lies Preface from 1893 Thanks those who helped pave his way: 1)Course with Sylow on Galois theory (1863) 2)Clebsch, Cremona, Klein, Adolf Mayer, and especially Camille Jordan 3)Darboux for promoting his geometrical work 4)Picard, first to recognize importance of Lies group theory for analysis 5)J. Tannery for sending students from ENS 6)Engel and Scheffers for writing his books

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Lie on Poincarés 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.)

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Kleins 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):

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Kleins 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 Lies earlier works

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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 Kleins 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

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Lie on Klein and the Erlangen Program from 1872 Words that Scandalized the German Mathematical Community

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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.

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Moreover, one finds hardly a trace of the all important concept of differential invariant in Kleins 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.

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Lie felt compelled to clarify these matters because Kleins pupils and friends have continually represented the relationship between Kleins works and mine falsely, and also because some of Kleins 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 Kleins 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.

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Seeking New Allies These remarks scandalized many within Kleins 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

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