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Algebraic Formalism within the Works of Servois and Its Influence on the Development of Linear Operator Theory - Introduction

Anthony J. Del Latto (Adelphi University) and Salvatore J. Petrilli, Jr. (Adelphi University)

Before the nineteenth century, algebra usually referred to the theory of solving equations. However, the field of algebra experienced an extensive transformation during the nineteenth century, a time period referred to by many historians as the Golden Age of mathematics. Consequently, by 1900 algebra encompassed the study of algebraic structures. One contributor to the advancement of algebra was François-Joseph Servois (1767-1847). Servois was a priest, artillery officer, professor of mathematics, and museum curator. Not only did he battle to defend Paris in 1814, but he also fought for an algebraic foundation for calculus. As we will see, Servois was an advocate of “algebraic formalism,” and the majority of his contributions to the field of mathematics fall under this category. In an “Essai” written in 1814, Servois attempted to provide a rigorous foundation for the calculus by introducing several algebraic properties, such as “commutativity” and “distributivity.” Essentially, he presented the notion of a field, an idea far ahead of his time. Although Servois was not successful in providing calculus with a proper foundation, his work did have an impact on the field of algebra, and influenced several mathematicians, including the English mathematicians Duncan Gregory and Robert Murphy. Many English mathematicians of this period used the works of French mathematicians to aid in their development of linear operator theory and abstract algebra [Koppleman 1971].

This article further illustrates that while the disciplines of algebra and analysis are studied separately today, mathematicians of the eighteenth century (and before) made little, if any, distinction between them. Finally, instructors can use the original sources found in this article to demonstrate to students the connection between classical and modern day mathematics.