# Algebraic Formalism within the Works of Servois and Its Influence on the Development of Linear Operator Theory - Servois' Belief in Algebraic Formalism

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

During the years 1811 to 1817, the majority of Servois' works were published in Joseph Diaz Gergonne's (1771-1859) Annales des mathématiques pures et appliquées. Much of his work focused on what Taton [1972a and 1972b] called “algebraic formalism." In 1814, we witness Servois' first defense of algebraic formalism, when he began a heated debate with Jean Robert Argand (1768-1822) and Jacques Français (1775-1833). In 1813, Français published a paper based on the work of Argand, in which he viewed complex numbers geometrically. In modern day mathematics, the view of complex numbers in the plane is known as the Argand Plane. Servois highly criticized the work of these two mathematicians, saying: “I had long thought of calling the ideas of Messrs. Argand and Français on complex numbers by the odious qualifications of useless and erroneous ....” [Servois 1814b, p. 228]. For instance, Servois argued against Français’ geometric “demonstration,” given in the preceding issue of Gergonne’s journal [1813], that the quantity $$a\sqrt{-1}$$ can be seen as the geometric mean of $$-a$$ and $$a$$. (In the field of complex numbers, we define the geometric mean of two real numbers $$a$$ and $$b$$ as $$\sqrt{ab}$$.) Furthermore, he went on to state that it was in the best interest of the science to express his personal view, because in this work he saw nothing but “a geometric mask applied to analytic forms ...." [Servois 1814b, pp. 228-230].

Servois' fight for “algebraic formalism” continued in 1814 with the publication of his “Essai sur un nouveau mode d'exposition des principes du calcul différential” [Servois 1814a] (“Essay on a New Method of Exposition of the Principles of Differential Calculus”). This work was an extension of Lagrange's research on the foundations of the differential calculus. In his “Essai,” Servois stated his belief that the differential calculus could be unified through algebraic generality:

In the preceding article, we have sketched the set of laws that brings together and unites all the differential functions, that is, the most general theory of the differential calculus. The practice of this calculus, which is nothing other than the execution of the operations given in the definitions .... [Servois 1814a, p. 122].

The notion of algebraic generality is apparent in the opening sections of his “Essai,” where Servois essentially defined a field for his set of functions under the operations of addition and composition [Bradley and Petrilli 2010]. However, it was not Servois' intention to create formal structures within algebra, but rather he “was concerned above all else to preserve the rigor and purity of algebra” [Taton 1972a].

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 - Servois' Belief in Algebraic Formalism," Loci (January 2012), DOI:10.4169/loci003802