Frédéric Brechenmacher, "Algebraic generality vs arithmetic generality in the 1874 controversy between C. Jordan and L. Kronecker," in  K. Chemla, R. Chorlay, D. Rabouin (eds.), The Oxford Handbook of Generality in Mathematics and the Sciences, Oxford University Press, 2016

 Book description : K. Chemla, R. Chorlay, D. Rabouin (eds.), The Oxford Handbook of Generality in Mathematics and the Science

Generality is a key value in scientific discourses and practices. Throughout history, it has received a variety of meanings and of uses. This collection of original essays aims to inquire into this diversity. Through case studies taken from the history of mathematics, physics and the life sciences, the book provides evidence of different ways of understanding the general in various contexts. It aims at showing how collectives have valued generality and how they have worked with specific types of "general" entities, procedures, and arguments.

The books connects history and philosophy of mathematics and the sciences at the intersection of two of the most fruitful contemporary lines of research: historical epistemology, in which values (e.g. "objectivity", "accuracy") are studied from a historical viewpoint; and the philosophy of scientific practice, in which conceptual developments are seen as embedded in networks of social, instrumental, and textual practices. Each chapter provides a self-contained case-study, with a clear exposition of the scientific content at stake. The collection covers a wide range of scientific domains - with an emphasis on mathematics - and historical periods. It thus allows a comparative perspective which suggests a non-linear pattern for a history of generality. The introductory chapter spells out the key issues and points to the connections between the chapters.

Chapter abstract : Frédéric Brechenmacher, "Algebraic generality vs arithmetic generality in the 1874 controversy between C. Jordan and L. Kronecker"

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.