Article ID: CBB000930655

La controverse de 1874 entre Camille Jordan et Leopold Kronecker (2007)

unapi

The 1874 Controversy between Camille Jordan and Leopold Kronecker During the whole of 1874, Camille Jordan and Leopold Kronecker quarrelled vigorously over the organisation of the theory of bilinear forms. That theory promised a general and homogeneous treatment of numerous questions arising in various 19th-century theoretical contexts, and it hinged on two theorems, stated independently by Jordan and Weierstrass, that would today be considered equivalent. It was, however, the perceived difference between those two theorems that sparked the 1874 controversy. Focusing on this quarrel allows us to explore the algebraic identity of the polynomial practices of the manipulations of forms in use before the advent of structural approaches to linear algebra. The latter approaches identified these practices with methods for the classification of similar matrices. We show that the practices---Jordan's canonical reduction and Kronecker's invariant computation---reflect identities inseparable from the social context of the time. Moreover, these practices reveal not only tacit knowledge, local ways of thinking, but also---in light of a long history tracing back to the work of Lagrange, Laplace, Cauchy, and Hermite---two internal philosophies regarding the significance of generality which are inseparable from two disciplinary ideals opposing algebra and arithmetic. By interrogating the cultural identities of such practices, this study aims at a deeper understanding of the history of linear algebra without focusing on issues related to the origins of theories or structures. Keywords : Algebraic practices. Generality. Jordan. Kronecker. Weierstrass. Canonical forms. Invariants; Elementary divisors. Invariant factors. Linear substitutions. Bilinear forms. Matrices. Spectral theory. Linear algebra. Algebra. Arithmetic. Mechanics.

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Description On their quarrels over the organization of the theory of bilinear forms, and “the algebraic identity of the polynomial practices of the manipulations of forms.” (from the abstract)


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Authors & Contributors
Boniface, Jacqueline
Boucard, Jenny
Kieldsen, Tinne Hoff
Gauthier, Yvon
Radu, Mircea
Schappacher, Norbert
Journals
Archive for History of Exact Sciences
Historia Mathematica
Revue d'Histoire des Mathématiques
Osmanli Bilimi Arastirmalari: Studies in Ottoman Science
Archives Internationales d'Histoire des Sciences
Journal of Dialectics of Nature
Publishers
Kluwer Academic
Ellipses
Open Court
American Mathematical Society
Birkhäuser
Crítica
Concepts
Mathematics
Arithmetic
Algebra
Linear algebra
Number theory; number concept
Logic
People
Kronecker, Leopold
Cauchy, Augustin Louis
Hilbert, David
Hölder, Ludwig Otto
Grassmann, Robert
Bolzano, Bernard
Time Periods
19th century
20th century, early
20th century
18th century
16th century
Places
France
Turkey
Italy
Germany
Great Britain
Institutions
University of Chicago
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