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Register a new userAutour de pratiques algebriques de Poincare : heritages de la reduction de Jordan
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Authors
Frederic Brechenmacher
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The legacy of Jordans canonical form on Poincares algebraic practices. This paper proposes a transversal overview on Henri Poincares early works (1878-1885). Our investigations start with a case study of a short note published by Poincare on 1884 : Sur les nombres complexes. In the perspective of todays mathematical disciplines - especially linear algebra -, this note seems completely isolated in Poincares works. This short paper actually exemplifies that the categories used today for describing some collective organizations of knowledge fail to grasp both the collective dimensions and individual specificity of Poincares work. It also highlights the crucial and transversal role played in Poincares works by a specific algebraic practice of classification of linear groups by reducing the analytical representation of linear substitution to their Jordans canonical forms. We then analyze in detail this algebraic practice as well as the roles it plays in Poincares works. We first provide a micro-historical analysis of Poincares appropriation of Jordans approach to linear groups through the prism of the legacy of Hermites works on algebraic forms between 1879 and 1881. This mixed legacy illuminates the interrelations between all the papers published by Poincare between 1878 and 1885 ; especially between some researches on algebraic forms and the development of the theory of Fuchsian functions. Moreover, our investigation sheds new light on how the notion of group came to play a key role in Poincares approach. The present paper also offers a historical account of the statement by Jordan of his canonical form theorem. Further, we analyze how Poincare transformed this theorem by appealing to Hermites
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During the whole of 1874, Camille Jordan and Leopold Kronecker quar- relled 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 prac- tices -- Jordans canonical reduction and Kroneckers 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, Cau- chy, 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.
Poincares approach to the three body problem has often been celebrated as a starting point of chaos theory in relation to the investigation of dynamical systems. Yet, Poincares strategy can also be analyzed as molded on - or casted in - some specific algebraic practices for manipulating systems of linear equations. These practices shed new light on both the novelty and the collective dimensions of Poincares Methodes nouvelles. As the structure of a cast-iron building may be less noticeable than its creative fac{c}ade, the algebraic cast of Poincares strategy is broken out of the mold in generating the novel methods of celestial mechanics. But as the various components that are mixed in some casting process can still be detected in the resulting alloy, the algebraic cast of the Methodes nouvelles points to some collective dimensions of Poincares methods. An edited version of the present preprint is to be published in the journal textit{Lastronomie} under the title Lapproche de Poincar`E sur le problEme des trois corps. This publication is an abstract in French language of a forthcoming paper - The algebraic cast of Poincar`Es textit{M`Ethodes nouvelles} - which will develop its main claims as well as the historiographical and mathematical issues raised in section 4 and section 5.
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