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

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

Although Servois was not successful in providing calculus with a proper foundation, his work did have an influence on the field of algebra, where he was a pioneer ahead of his time. He knew that, when performing algebraic manipulations on quantities, he needed to have a structure consisting of a set that obeyed certain axioms. Additionally, his work on analysis spread to England and significantly influenced mathematicians such as Duncan Gregory and Robert Murphy in their efforts to establish the foundations of linear operator theory. Servois' main contribution to this development was recognizing the “distributive” and “commutative” properties of operators, terms that he coined, and the method of separating symbols and their operations. Koppleman [1971] stated that the English mathematicians found the tools for the calculus of operations in the works of French mathematicians, such as Servois'. She went on further to say that “it was the English who developed this work in the calculus of operations both in extending the scope of its applications and in relating it to the theory of abstract algebra” [Koppleman 1971, p. 175].

The material discussed in this paper can aid teachers and students of abstract algebra, linear algebra, and the history of mathematics. By presenting the history of mathematics, instructors can illustrate the idea that mathematics is a constantly evolving field. Besides providing a readable account of the history of algebraic structures and the beginnings of linear operator theory, this paper contains many explanations of and references to original sources. Additionally, this article highlights the fact 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. When analyzing the original sources, the student co-author of this article, Anthony, was initially somewhat shocked to see so many calculus-like and algebraic ideas presented together. As an undergraduate, he had rarely seen ideas from both analysis and algebra meshed so closely together.

In the spirit of Victor Katz, we would encourage instructors to incorporate original sources within their classrooms. This paper provides references to original sources that can easily be found on the internet. With a little consideration on the part of the instructor, it is easy to create historical activities that can fit in any mathematics course. For instance, pages 1-13 of Gregory's The Mathematical Writings contain many examples of symbolical algebra that can be incorporated into any course, such as a first-year calculus course. An instructor could provide a copy of Gregory's discussion of operators that satisfy the property $f(x) + f(y) = f(xy)$ and ask students to write a list of functions that satisfy this property.

Furthermore, these sources provide students with an opportunity to conduct research on the history of mathematics. Open questions include, for instance:

• Who was Thomas Jarrett? Did he receive any formal mathematical training? If so, from whom did he receive training? Are there any other mathematical works published by him?
• Jarrett's work is taken from a few mathematicians [Jarrett 1831, pp. III-V]. Many are well-known, but who was Ferdinand Franz Schwiens? Other than the analysis textbook mentioned by Jarrett, what did Schwiens write?
• Many unanswered questions still remain regarding Servois' mathematical career. For instance, was there correspondence between Servois and any English mathematicians? If so, it would be interesting to use it to explore the extent of Servois' influence on these mathematicians.

Anthony J. Del Latto (Adelphi University) and Salvatore J. Petrilli, Jr. (Adelphi University), "Algebraic Formalism within the Works of Servois and Its Influence on the Development of Linear Operator Theory - Conclusions and Recommendations," Convergence (January 2012), DOI:10.4169/loci003802