A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics The quarrel : a chronology Jordan. December 1873. "Sur les Polynômes bilinéaires". Note l’Académie des Sciences de Paris. Kronecker. December 22, 1873. "Ueber schaaren von quadratischen und bilinearen formen". Academy of Berlin. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics The quarrel : a chronology Jordan to Kronecker, December 1873 Je ne voudrais pas . . . , que je désire la guerre, et que je préférerais une polémique / guerre/ des dé- bats publics à des explications amicales. Ce n’est pas moi qui ait ouvert les hostilités commencé la polémique. J’ai publié il est vrai (c’était mon droit évident) sans vous consulter des recherches qui complétaient les vôtres [. . . ] Si au lieu de jeter brusquement ce débat dans le public, vous vous étiez adressé à moi [...] j’aurais constaté immédiatement, ce que j’ai reconnu trop tard, que votre méthode de 1868 relu plus attentive- ment votre mémoire de 1868 et constaté, ce que je n’avais pas remarqué à première vue, que les formes bilinéaires non citées dans votre travail, y sont pourtant implicitement comprises. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics The quarrel : a chronology December 1873 to March 1874 : private correspondence. mostly about the issue of priority / the collective dimensions of some texts published in Berlin. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics The quarrel : a chronology December 1873 to March 1874 : private correspondence. mostly about the issue of priority / the collective dimensions of some texts published in Berlin. A "levis culpa" (Kronecker’s own work in regard with Hermite’s and Serret’s) : Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics The quarrel goes public again Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics The quarrel goes public again Kronecker 1874 : 367 Should such general expressions be found, one should in every case be able to justify calling all of them canonical forms on the basis of their generality and simplicity ; but if one does not want to stick to the purely formal viewpoint which is often put the fore in the more recent Algebra – certainly not for the greatest advantage of the true knowledge -, one shall not omit to derive the correction of these canonical forms on the basis of inner grounds. Truly, these so called canonical or normal forms are determined only by the orientation of the study, but they should not be seen as the aim of the research. . . Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics The quarrel goes public again Kronecker. January – March 1874. "Ueber quadratische und bilineare Formen". Academy of Berlin. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics The quarrel goes public again Kronecker. January – March 1874. "Ueber quadratische und bilineare Formen". Academy of Berlin. Jordan. March 1874 . "Sur les formes bilinéaires". Publication in the Journal de Liouville of the memoir announced in 1873 Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics The quarrel goes public again Kronecker. January – March 1874. "Ueber quadratische und bilineare Formen". Academy of Berlin. Jordan. March 1874 . "Sur les formes bilinéaires". Publication in the Journal de Liouville of the memoir announced in 1873 Jordan. March, 2 1874. "Sur la réduction des formes bilinéaires". Academy of Paris. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics The quarrel goes public again Kronecker. January – March 1874. "Ueber quadratische und bilineare Formen". Academy of Berlin. Jordan. March 1874 . "Sur les formes bilinéaires". Publication in the Journal de Liouville of the memoir announced in 1873 Jordan. March, 2 1874. "Sur la réduction des formes bilinéaires". Academy of Paris. Kronecker. April 1874. "Sur les faisceaux de formes quadratiques et bilinéaires". Academy of Paris. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics The quarrel goes public again Kronecker. January – March 1874. "Ueber quadratische und bilineare Formen". Academy of Berlin. Jordan. March 1874 . "Sur les formes bilinéaires". Publication in the Journal de Liouville of the memoir announced in 1873 Jordan. March, 2 1874. "Sur la réduction des formes bilinéaires". Academy of Paris. Kronecker. April 1874. "Sur les faisceaux de formes quadratiques et bilinéaires". Academy of Paris. Kronecker. May 1874. "Nachtrag". Sequel to the memoir of March, Academy of Berlin. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics The quarrel goes public again Kronecker. January – March 1874. "Ueber quadratische und bilineare Formen". Academy of Berlin. Jordan. March 1874 . "Sur les formes bilinéaires". Publication in the Journal de Liouville of the memoir announced in 1873 Jordan. March, 2 1874. "Sur la réduction des formes bilinéaires". Academy of Paris. Kronecker. April 1874. "Sur les faisceaux de formes quadratiques et bilinéaires". Academy of Paris. Kronecker. May 1874. "Nachtrag". Sequel to the memoir of March, Academy of Berlin. Jordan. June 1874. "Sur les systèmes de formes quadratiques". Academy of Paris, Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics The quarrel goes public again Kronecker. January – March 1874. "Ueber quadratische und bilineare Formen". Academy of Berlin. Jordan. March 1874 . "Sur les formes bilinéaires". Publication in the Journal de Liouville of the memoir announced in 1873 Jordan. March, 2 1874. "Sur la réduction des formes bilinéaires". Academy of Paris. Kronecker. April 1874. "Sur les faisceaux de formes quadratiques et bilinéaires". Academy of Paris. Kronecker. May 1874. "Nachtrag". Sequel to the memoir of March, Academy of Berlin. Jordan. June 1874. "Sur les systèmes de formes quadratiques". Academy of Paris, Jordan. July 1874. "Mémoire sur la réduction et la transformation des systèmes quadratiques", Journal de Liouville Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics The quarrel goes public again Kronecker. January – March 1874. "Ueber quadratische und bilineare Formen". Academy of Berlin. Jordan. March 1874 . "Sur les formes bilinéaires". Publication in the Journal de Liouville of the memoir announced in 1873 Jordan. March, 2 1874. "Sur la réduction des formes bilinéaires". Academy of Paris. Kronecker. April 1874. "Sur les faisceaux de formes quadratiques et bilinéaires". Academy of Paris. Kronecker. May 1874. "Nachtrag". Sequel to the memoir of March, Academy of Berlin. Jordan. June 1874. "Sur les systèmes de formes quadratiques". Academy of Paris, Jordan. July 1874. "Mémoire sur la réduction et la transformation des systèmes quadratiques", Journal de Liouville Kronecker. 1874. "Uber die congruenten Transformationen der bilinearen formen". Academy of Berlin. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Much a do about nothing ? Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Much a do about nothing ? Two theorems equivalent from the standpoint of linear algebra... Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Much a do about nothing ? Two theorems equivalent from the standpoint of linear algebra... ... which did not exist as a discipline before the 1930s Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Much a do about nothing ? Two theorems equivalent from the standpoint of linear algebra... ... which did not exist as a discipline before the 1930s A controversy about the very nature of algebra and its role in mathemaics Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Much a do about nothing ? Two theorems equivalent from the standpoint of linear algebra... ... which did not exist as a discipline before the 1930s A controversy about the very nature of algebra and its role in mathemaics Jordan’s practice of canonical reduction vs Kronecker’s practice of invariant computation. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Much a do about nothing ? Two theorems equivalent from the standpoint of linear algebra... ... which did not exist as a discipline before the 1930s A controversy about the very nature of algebra and its role in mathemaics Jordan’s practice of canonical reduction vs Kronecker’s practice of invariant computation. The opposition between two mathematical cultures Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Much a do about nothing ? Two theorems equivalent from the standpoint of linear algebra... ... which did not exist as a discipline before the 1930s A controversy about the very nature of algebra and its role in mathemaics Jordan’s practice of canonical reduction vs Kronecker’s practice of invariant computation. The opposition between two mathematical cultures What were the collective dimensions of such cultures ? France vs Germany / an echo of the 1870 Prussian war ? Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Much a do about nothing ? Two theorems equivalent from the standpoint of linear algebra... ... which did not exist as a discipline before the 1930s A controversy about the very nature of algebra and its role in mathemaics Jordan’s practice of canonical reduction vs Kronecker’s practice of invariant computation. The opposition between two mathematical cultures What were the collective dimensions of such cultures ? France vs Germany / an echo of the 1870 Prussian war ? A shared algebraic culture at the start of the controversy Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics The reference to a shared algebraic culture Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics The reference to a shared algebraic culture Antoine Yvon-Villarceau, 1870, "Note sur les conditions des petites oscillations d’un corps solide de figure quelconque et la théorie des équations différentielles linéaires" Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics The reference to a shared algebraic culture Antoine Yvon-Villarceau, 1870, "Note sur les conditions des petites oscillations d’un corps solide de figure quelconque et la théorie des équations différentielles linéaires" Problems of small oscillations in Lagrange’s works about a hundred years before . Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics From the small oscillations of swinging strings to the ones of periodic trajectories Lagrange, 1766 : small oscillations ξ i ( t ) of a string loaded with n bodies Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics From the small oscillations of swinging strings to the ones of periodic trajectories Lagrange, 1766 : small oscillations ξ i ( t ) of a string loaded with n bodies by neglecting the non linear terms in the power series developments of the equations of dynamics Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics From the small oscillations of swinging strings to the ones of periodic trajectories Lagrange, 1766 : small oscillations ξ i ( t ) of a string loaded with n bodies by neglecting the non linear terms in the power series developments of the equations of dynamics In the 1770s, Lagrange and Laplace have transfered this mathematization to the investigation of the "secular inequalities in planetary theory" i.e. to the small oscillations of the planets of the solar system on their orbits. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics A system of n linear equations with constant coefficients : d ξ i � dt = A i , j ξ j j = 1 , n The integration of the system is based on its decomposition into n independent equations d ξ i dt = α i ξ j Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics A system of n linear equations with constant coefficients : d ξ i � dt = A i , j ξ j j = 1 , n The integration of the system is based on its decomposition into n independent equations d ξ i dt = α i ξ j a mathematization of the mechanical observation that the oscillations of a swinging string loaded with n bodies can be decomposed into the independent oscillations of n strings loaded with a single body (Bernouilli) Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics A system of n linear equations with constant coefficients : d ξ i � dt = A i , j ξ j j = 1 , n The integration of the system is based on its decomposition into n independent equations d ξ i dt = α i ξ j a mathematization of the mechanical observation that the oscillations of a swinging string loaded with n bodies can be decomposed into the independent oscillations of n strings loaded with a single body (Bernouilli) Let S be the periodicity of such a proper oscillation (i.e. an eigenvalue of A − SI ) : Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics A system of n linear equations with constant coefficients : d ξ i � dt = A i , j ξ j j = 1 , n The integration of the system is based on its decomposition into n independent equations d ξ i dt = α i ξ j a mathematization of the mechanical observation that the oscillations of a swinging string loaded with n bodies can be decomposed into the independent oscillations of n strings loaded with a single body (Bernouilli) Let S be the periodicity of such a proper oscillation (i.e. an eigenvalue of A − SI ) : Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics A system of n linear equations with constant coefficients : d ξ i � dt = A i , j ξ j j = 1 , n The integration of the system is based on its decomposition into n independent equations d ξ i dt = α i ξ j a mathematization of the mechanical observation that the oscillations of a swinging string loaded with n bodies can be decomposed into the independent oscillations of n strings loaded with a single body (Bernouilli) Let S be the periodicity of such a proper oscillation (i.e. an eigenvalue of A − SI ) : � � A 1 , 1 − S A 1 , 2 ... A 1 , n � � � � A 2 , 1 A 2 , 2 − S ... A 2 , n � � = 0 � � ... ... ... ... � � � � A n , 1 A n , 2 ... A n , n − S � � Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics � � A 1 , 1 − S A 1 , 2 ... A 1 , n � � � � A 2 , 1 A 2 , 2 − S A 2 , n ... � � = 0 � � ... ... ... ... � � � � A n , 1 A n , 2 ... A n , n − S � � An algebraic equation of degree n : "the equation to the secular inequalities in planetary theory" (the secular equation for short) ξ i ( t ) = C 1 e α 1 t + C 2 e α 2 t + ... + C n e α n t Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics � � A 1 , 1 − S A 1 , 2 ... A 1 , n � � � � A 2 , 1 A 2 , 2 − S A 2 , n ... � � = 0 � � ... ... ... ... � � � � A n , 1 A n , 2 ... A n , n − S � � An algebraic equation of degree n : "the equation to the secular inequalities in planetary theory" (the secular equation for short) to each root α i of this equation : a proper oscillation ξ i ( t ) = e α i t ξ i ( t ) = C 1 e α 1 t + C 2 e α 2 t + ... + C n e α n t Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics � � A 1 , 1 − S A 1 , 2 ... A 1 , n � � � � A 2 , 1 A 2 , 2 − S A 2 , n ... � � = 0 � � ... ... ... ... � � � � A n , 1 A n , 2 ... A n , n − S � � An algebraic equation of degree n : "the equation to the secular inequalities in planetary theory" (the secular equation for short) to each root α i of this equation : a proper oscillation ξ i ( t ) = e α i t If the equation has n distinct roots, one thus gets n independent solutions ξ i ( t ) = C 1 e α 1 t + C 2 e α 2 t + ... + C n e α n t Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Lagrange’s algebraic procedure for manipulating linear systems A rational expression of the coordinates ( x α j i ) of the solutions of symetric linear systems of n equations with constant coefficients. = ∆ 1 i x α j ( α j ) i ∆ S − α j Involving ∆( S ) , the (polynomial) characteristic determinant of the system A , i.e. the one that generates the secular equation : det ( A − SI ) its (polynomial) successive minors ∆ 1 i ( S ) (developments / first line and ith column) Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics This expression is provided by the non-zero column of the cofactor matrix of A − SI . For example, given � 1 − 1 0 � � � � − 1 2 1 � A = � � � � 0 1 1 � � The characteristic equation : det ( A − SI ) = ∆( S ) = S ( 3 − S )( 1 − S ) ∆ 11 ( S ) = ( 1 − S )( 2 − S ) − 1, ∆ 12 ( S ) = ( 1 − S ) , ∆ 13 ( S ) = 1 e.g. for the eigenvalue s 1 = 1, the coordinates of an eigenvector are : ( 1 ) = 1 ( 1 ) = − 1 1 = ∆ 11 2 = ∆ 12 3 = ∆ 13 x s 1 2 , x s 1 ( 1 ) = 0 , x s 1 ∆ ∆ ∆ 2 s − 1 s − 1 s − 1 Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics The problem of multiple roots = ∆ 1 i x α j ( α j ) i ∆ S − α j Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Mechanical stability and algebraic multiplicity Lagrange’s 1866 criterion of mechanical stability in function of the algebraic nature of the roots of the secular equation : The mechanical system is stable iff the α i are real, negatives and distinct . In this situation a particular solution has the form sin ( α i t ) Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Mechanical stability and algebraic multiplicity Lagrange’s 1866 criterion of mechanical stability in function of the algebraic nature of the roots of the secular equation : The mechanical system is stable iff the α i are real, negatives and distinct . In this situation a particular solution has the form sin ( α i t ) in the case of imaginary roots : some exponential oscillations Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Mechanical stability and algebraic multiplicity Lagrange’s 1866 criterion of mechanical stability in function of the algebraic nature of the roots of the secular equation : The mechanical system is stable iff the α i are real, negatives and distinct . In this situation a particular solution has the form sin ( α i t ) in the case of imaginary roots : some exponential oscillations in the case of a multiple root : tsin ( α i t ) : "le temps sort du sinus" and generates some non periodic unbounded oscillations (false :Weierstrass 1858, Jordan 1871) Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Mechanical stability and algebraic multiplicity Antoine Yvon-Villarceau, 1870, I claim that this condition is not necessary for the oscillations to remain small. . . . Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Jordan’s response to Villarceau in 1871 dx 1 dt = a 1 x 1 + ... + l 1 x n dx 2 dt = a 2 x 1 + ... + l 2 x n ... dx n dt = a n x 1 + ... + l n x n Ce problème peut se résoudre très simplement par un procédé identique à celui dont nous nous sommes servi, dans notre Traité des substitutions , pour ramener une substitution linéaire quelconque à sa forme canonique. dy 1 dt = σ y 1 , dz 1 dt = σ z 1 + y , du 1 dt = σ u 1 + z 1 , ..., dw 1 = σ w 1 + v 1 dt (...) w 1 = e σ t ψ ( t ) , v 1 = e σ t ψ ′ ( t ) , ..., y 1 = e σ t ψ r ( t ) , ψ ( t ) étant une fonction entière arbitraire du degré r − 1. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Jordan 1871 : first response to Villarceau / application of the canonical form theorem to linear differential equations with constant coefficients Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Jordan 1871 : first response to Villarceau / application of the canonical form theorem to linear differential equations with constant coefficients Jordan 1872 : second response to Villarceau : in the case of mechanics, systems can always be reduced to a diagonal form / pairs of quadratic forms Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Jordan 1871 : first response to Villarceau / application of the canonical form theorem to linear differential equations with constant coefficients Jordan 1872 : second response to Villarceau : in the case of mechanics, systems can always be reduced to a diagonal form / pairs of quadratic forms A result already stated by Weierstrass in 1858 in discussing Lagrange’s criterion of stability : multiplicity of roots do not interfere with stability Jordan 1873 : the non symetric case of pairs of bilinear forms Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Jordan 1871 : first response to Villarceau / application of the canonical form theorem to linear differential equations with constant coefficients Jordan 1872 : second response to Villarceau : in the case of mechanics, systems can always be reduced to a diagonal form / pairs of quadratic forms A result already stated by Weierstrass in 1858 in discussing Lagrange’s criterion of stability : multiplicity of roots do not interfere with stability Jordan 1873 : the non symetric case of pairs of bilinear forms A result already stated by Weierstrass in 1868 Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Two ends given to a shared history The 1874 controversy : opposing two ends given to a shared history, implicitly referring to the works of d’Alembert Lagrange, Laplace, Cauchy etc. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Two ends given to a shared history The 1874 controversy : opposing two ends given to a shared history, implicitly referring to the works of d’Alembert Lagrange, Laplace, Cauchy etc. It was because of this, that some identities between Jordan’s and Weierstrass’ theorems had arisen between 1870 and 1873. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Two ends given to a shared history The 1874 controversy : opposing two ends given to a shared history, implicitly referring to the works of d’Alembert Lagrange, Laplace, Cauchy etc. It was because of this, that some identities between Jordan’s and Weierstrass’ theorems had arisen between 1870 and 1873. It was also in reference to this history that the quarrel developped Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Kronecker’s views on the history of generality in algebra Kronecker 1874a : 367 One is indeed used to discovering essentially new difficulties – especially in algebraic questions -, as soon as ... one forces his way through the surface of this so called generality - which excludes any particularity-, one penetrates the true generality - which encompasses all singularities-, one generally finds the real difficulties of the study, but at the same time one finds the wealth of new viewpoints and phenomena which lie in its depths. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Kronecker’s views on the history of generality in algebra Kronecker 1874a : 367 This holds in the few algebraic questions which have been tackled completely and to the smallest details, such as the theory of networks of quadratic forms ... As long as one did not dare to dispense with the hypothesis that the determinant has only unequal factors, one can reach only inadequate results in the well known problem ... which has been dealt with so often over the last century - ; under this hypothesis, the true viewpoints on the investigation remained completely unacknowledged. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Kronecker’s views on the history of generality in algebra Kronecker 1874a : 367 Weierstrass’ 1858 work dropped this hypothesis ... the general introduction of the notion of elementary divisor ... the brightest light was shed on the new algebraic configurations, and at the same time by this complete treatment of the subject, the most valuable insights were reached on the theory of the higher invariants, as conceived in their true generality. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics A shared algebraic culture ; the secular equation Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics A shared algebraic culture ; the secular equation Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics The circulation of Lagrange’s algebraic procedure Cauchy, 1829, Sur l’équation à l’aide de laquelle on détermine les inégalités séculaires des planètes The problem of finding the principal axis of a conic (with real coefficients) f ( x 1 , x 2 , ..., x n ) = A 11 x 2 1 + A 22 x 2 2 + ... + A nn x 2 n + 2 A 12 x 1 x 2 + 2 A 13 x 1 x 3 + ... To transform it into a sum of squares : "the secular equation" Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics The circulation of Lagrange’s algebraic procedure Cauchy, 1829, Sur l’équation à l’aide de laquelle on détermine les inégalités séculaires des planètes The problem of finding the principal axis of a conic (with real coefficients) f ( x 1 , x 2 , ..., x n ) = A 11 x 2 1 + A 22 x 2 2 + ... + A nn x 2 n + 2 A 12 x 1 x 2 + 2 A 13 x 1 x 3 + ... To transform it into a sum of squares : "the secular equation" 1 + ∆ n − 2 2 + ... + ∆ f ( x 1 , x 2 , ..., x n ) = ∆ n − 1 X 2 X 2 X 2 n ∆ n − 1 ∆ 1 with coefficients given by the successive principal minors of the polynomial determinant ∆( S ) Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics The circulation of a specific problem : multiple roots In both expressions : = ∆ 1 i x α j ( α j ) i ∆ S − α j 1 + ∆ n − 2 2 + ... + ∆ f ( x 1 , x 2 , ..., x n ) = ∆ n − 1 X 2 X 2 X 2 n ∆ n − 1 ∆ 1 Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics The circulation of a specific problem : multiple roots A problem similar to the one of the root of a negative number / the development of complex analysis : Cauchy’s Residue theory Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics The circulation of a specific problem : multiple roots A problem similar to the one of the root of a negative number / the development of complex analysis : Cauchy’s Residue theory In the 1850s, different algebraic approaches to the problem of the multiplicity of roots : Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics The circulation of a specific problem : multiple roots A problem similar to the one of the root of a negative number / the development of complex analysis : Cauchy’s Residue theory In the 1850s, different algebraic approaches to the problem of the multiplicity of roots : Charles Hermite’s algebraic theory of quadratic forms Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics The circulation of a specific problem : multiple roots A problem similar to the one of the root of a negative number / the development of complex analysis : Cauchy’s Residue theory In the 1850s, different algebraic approaches to the problem of the multiplicity of roots : Charles Hermite’s algebraic theory of quadratic forms James Joseph Sylvester’s notions of matrices and minors Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics The circulation of a specific problem : multiple roots A problem similar to the one of the root of a negative number / the development of complex analysis : Cauchy’s Residue theory In the 1850s, different algebraic approaches to the problem of the multiplicity of roots : Charles Hermite’s algebraic theory of quadratic forms James Joseph Sylvester’s notions of matrices and minors Karl Weierstrass’s elementary divisors theorem Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics The circulation of a specific problem : multiple roots A problem similar to the one of the root of a negative number / the development of complex analysis : Cauchy’s Residue theory In the 1850s, different algebraic approaches to the problem of the multiplicity of roots : Charles Hermite’s algebraic theory of quadratic forms James Joseph Sylvester’s notions of matrices and minors Karl Weierstrass’s elementary divisors theorem Camille Jordan’s canonical form in finite groups theory Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics The circulation of a specific problem : multiple roots A problem similar to the one of the root of a negative number / the development of complex analysis : Cauchy’s Residue theory In the 1850s, different algebraic approaches to the problem of the multiplicity of roots : Charles Hermite’s algebraic theory of quadratic forms James Joseph Sylvester’s notions of matrices and minors Karl Weierstrass’s elementary divisors theorem Camille Jordan’s canonical form in finite groups theory Different lines of developments : a strong structuration of the algebraic methods used at the end of the 19th century Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics The circulation of a specific problem : multiple roots A problem similar to the one of the root of a negative number / the development of complex analysis : Cauchy’s Residue theory In the 1850s, different algebraic approaches to the problem of the multiplicity of roots : Charles Hermite’s algebraic theory of quadratic forms James Joseph Sylvester’s notions of matrices and minors Karl Weierstrass’s elementary divisors theorem Camille Jordan’s canonical form in finite groups theory Different lines of developments : a strong structuration of the algebraic methods used at the end of the 19th century The 1874 controversy : two attempts to give new theoretical identities to wat used to be a broadly shared algebraic culture. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Hermite’s algebraic theory of forms In the 1850s, Hermite and Sylvester have looked for a purely algebraic proof of Sturm theorem through the investigation of the specific case of the secular equation (Sinaceur 1991) Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Hermite’s approach to Sturm theorem To the secular equation one can associate a quadratic form : f ( x 1 , x 2 , ..., x n ) = A 11 x 2 1 + A 22 x 2 2 + ... + A nn x 2 n + 2 A 12 x 1 x 2 + 2 A 13 x 1 x 3 + ... which can be transformed into a sum of squares: 1 + ∆ n − 2 2 + ... + ∆ f ( x 1 , x 2 , ..., x n ) = ∆ n − 1 X 2 X 2 X 2 n ∆ n − 1 ∆ 1 Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Hermite’s approach to Sturm theorem To the secular equation one can associate a quadratic form : f ( x 1 , x 2 , ..., x n ) = A 11 x 2 1 + A 22 x 2 2 + ... + A nn x 2 n + 2 A 12 x 1 x 2 + 2 A 13 x 1 x 3 + ... which can be transformed into a sum of squares: 1 + ∆ n − 2 2 + ... + ∆ f ( x 1 , x 2 , ..., x n ) = ∆ n − 1 X 2 X 2 X 2 n ∆ n − 1 ∆ 1 The number of positive and negative signs is an invariant of the quadratic form (Sylvester’s inertia law) that actually provides the number of real distinct roots of the secular equation (and more generally an algebraic proof of Sturm theorem). Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Hermite’s theory of forms "The arithmetic theory of quadratic forms" : in Gauss’s legacy : operations by substitutions with integer coefficients Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Hermite’s theory of forms "The arithmetic theory of quadratic forms" : in Gauss’s legacy : operations by substitutions with integer coefficients "The algebraic theory of forms" : operations by substitutions with real coefficients / algebraic because its role in Sturm theorem Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Hermite’s theory of forms "The arithmetic theory of quadratic forms" : in Gauss’s legacy : operations by substitutions with integer coefficients "The algebraic theory of forms" : operations by substitutions with real coefficients / algebraic because its role in Sturm theorem Kronecker 1873, "Sur la theorie algébrique des formes quadratiques" Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Two ends given to a shared history A shared algebraic culture The algebraic theory of forms Algebra vs arithmetics Hermite’s theory of forms "The arithmetic theory of quadratic forms" : in Gauss’s legacy : operations by substitutions with integer coefficients "The algebraic theory of forms" : operations by substitutions with real coefficients / algebraic because its role in Sturm theorem Kronecker 1873, "Sur la theorie algébrique des formes quadratiques" Darboux 1874, "Sur la theorie algébrique des formes quadratiques" Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Kronecker : arithmetics A shared algebraic culture Jordan : algebra Algebra vs arithmetics Kronecker : the arithmetic theory of forms / invariant factors Kronecker 1874b : 415 In the arithmetical theory of forms, one must certainly be satisfied by the indication of a procedure for deciding of the question of the equivalence, and this problem was indeed formulated explicitly in this way too (cf. Gauss : Disquitiones arithmeticae, Sectio V.). The procedure itself is here also based on the transformation to reduced forms: but it must not be forgotten that, in the arithmetic theory, these [reduced forms] have a completely different meaning than the one they have in the Algebra. Indeed, there, the invariants... can be directly defined, although not explicitly but only described as the final result of arithmetic operations ; for much the same is true with most concepts of arithmetic, e.g. even the simple notion greatest common divisor. Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Kronecker : arithmetics A shared algebraic culture Jordan : algebra Algebra vs arithmetics Jordan and the generality of algebra : simplicity Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
A quarrel of priority ? Kronecker : arithmetics A shared algebraic culture Jordan : algebra Algebra vs arithmetics Jordan : simplicity / an "algebraic process of reduction" Jordan’s specific approach to group theory and Galois theory since his 1860 thesis : Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker
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