The Golden Age of Analysis

A key feature to understanding the history of a mathematical field is becoming acquainted with the progression of its development. The beginning of the nineteenth century was considered the Golden Age of mathematics, a time when many fields, such as abstract algebra and geometry, were given a rigorous foundation [Piccolino 1984]. Another area of mathematics that experienced fundamental change was analysis. The turning point for rigor in calculus might have been the publication of Théorie des fonctions analytiques [Lagrange 1797], in which Joseph-Louis Lagrange argued that calculus ought to be placed on a foundation of algebraic analysis. In fact, Lagrange gave this book the following subtitle “The Principles of the differential Calculus, freed from any consideration of the infinitely small or vanishing quantities, of limits or of fluxions, and reduced to the algebraic Analysis of finite quantities.”

Figure 1. Portrait of Joseph-Louis Lagrange (public domain).

During the early nineteenth century, mathematicians began to give a rigorous foundation to calculus; however, rigor does not have a definitive definition. Judith Grabiner described this undertaking as follows:

First, every concept of the subject had to be explicitly defined in terms of concepts whose nature was held to the already known …. Second, theorems had to be proved, with every step in the proof justified by a previously proved theorem, by a definition, or by an explicitly stated axiom. Third, the definitions chosen, and the theorems proved, had to be sufficiently broad to support the entire structure of valid results belonging to the subject [Grabiner 1981, p. 5].