Fragments of a History of the Concept of Ideal. Poncelet's and Chasles's Reflections on Generality in Geometry and their Impact on Kummer's Work with Ideal Divisors

TL;DR

This paper explores how 19th-century geometric ideas about ideals influenced Kummer's number theory, highlighting the philosophical and mathematical connections that shaped the development of ideal divisors.

Contribution

It demonstrates the historical and conceptual link between geometric ideals and algebraic ideal divisors, emphasizing the influence of French geometric philosophy on Kummer's work.

Findings

Kummer's ideal prime factors were inspired by geometric ideals.

Ideal elements in geometry influenced the development of ideal divisors in number theory.

Philosophical reflections on generality contributed to the evolution of projective geometry.

Abstract

In this essay, I argue for the following theses. First, Kummer's concept of ''ideal prime factors of a complex number'' was inspired by Poncelet's introduction of ideal elements in geometry as well as by the reconceptualization that Michel Chasles put forward for them in 1837. In other words, the idea of ideal divisors in Kummer's ''theory of complex numbers'' derives from the introduction of ideal elements in the new geometry. This is where the term ''ideal'' comes from. Second, the introduction of ideal elements into geometry and the subsequent reconceptualization of what was in play with these elements were linked to philosophical reflections on generality that practitioners of geometry in France developed in the first half of the 19${}^{\rm th}$ century in order to devise a new approach to geometry, which would eventually become projective geometry. These philosophical reflections circulated as such and played a key part in the advance of other domains, including in Kummer's major innovation in the context of number theory.