Algebraic generality vs arithmetic generality in the controversy between C. Jordan and L. Kronecker (1874)

TL;DR

The paper examines the 1874 controversy between Jordan and Kronecker over different mathematical practices, highlighting cultural and philosophical differences in their approaches to generality and mathematical identity.

Contribution

It analyzes the historical and cultural context of the Jordan-Kronecker quarrel, revealing how practices and perceptions shaped mathematical concepts of generality.

Findings

Revealed cultural differences in mathematical practices

Linked historical context to conceptual disagreements

Highlighted the role of tacit knowledge in mathematical development

Abstract

Throughout the whole year of 1874, C. Jordan and L. Kronecker were quarrelling over two theorems. On the one hand, Jordan had stated in 1870 a canonical form theorem for substitutions of linear groups; on the other hand, Karl Weierstrass had introduced in 1868 the elementary divisors of non singular pairs of bilinear forms (P,Q) in stating a key theorem of the theory of bilinear and quadratic forms. Although they would be considered equivalent as regard to modern mathematics, not only had these two theorems been stated independently and for different purposes, they had also been lying within the distinct frameworks of two theories until some connections came to light in 1872-1873, breeding the 1874 quarrel and hence revealing an opposition over two practices relating to distinctive cultural features. As we will be looking into the 1874 quarrel, our purpose will be to show how the complex identities of practices such as Jordan s canonical reduction and Kronecker s invariant computation highlight some cultural issues such as tacit knowledge and perceptions of history peculiar to individuals or communities as well as some local ways of thinking such as disciplinary ideals and internal philosophies of generality and simplicity.