Algebraic generality vs arithmetic generality in the controversy between C. Jordan and L. Kronecker (1874).
Résumé
Throughout the whole year of 1874, Camille Jordan and Leopold Kronecker were quarrelling over two practices relative to two theorems. On the one hand, Weierstrass had defined in 1868 a complete set of polynomial invariants for the characterisation of non singular pairs of bilinear forms (P,Q) ; computed from the determinant |P+sQ|, they were designated as the elementary divisors of (P,Q). It was in the very different context of group theory that, on the other hand, Jordan had stated in 1870 that a linear substitution could be reduced to a "simple canonical form". The two theorems were stated independently, belonged to distinct theories and, although they would be considered equivalent as regard to modern mathematics, it was their opposition that generated the 1874 controversy. As we will be looking into this quarrel, our purpose will be to discuss the algebraic identities of practices used before the time of linear algebra when these practices would be seen as methods for the classification of similar matrices. We will see that the different ways the two opponents referred to the history of a hundred years old mechanical problem and the works of Lagrange, Laplace, Cauchy and Hermite shed some light on how the complex identities of practices such as Jordan's canonical reduction and Kronecker's invariant computation are related to an opposition of two disciplinary ideals and internal philosophies of "generality" in arithmetic and algebra.
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