Specialeforsvar ved Lisa Fríðgerð Danielsen

Title: The correlation between the notation and the formation of concept in the development of calculus from 1665 to 1900

Abstract:
The scope of this thesis was to investigate the influence of and the interaction between the creation of new notation and the formation of new concepts throughout the development of calculus from the calculus of Leibniz and Newton to the implementation of the Weierstrassian notation in the work of e.g. Heine and Dini in the late 19th century. This obiviously included the contributions to both notation and formation of concept found in the works of Leibniz, Euler, Cauchy and Weierstrass, as well as the philosophical aspect of the role of the symbols.

As philosophical theory I have used the theory of distributed cognition and in particular the extended mind thesis with the perspective on mathematics and mathematical notation. Hence this thesis is a study of a development, which begins in a geometrical setting, where curves were the central objects of investigation. Symbols and variables were associated to these curves in order to help with the calculations. Leibniz clever use of these symbols facilitated a development of the concept into equations with geometric entities, which was developed by Euler into a study of algebraic equation and functions.

Euler had a very liberate and unrestricted approach to his use of symbols, which resulted in the weird formal sums of divergent series. This caused a reaction from Cauchy, who began a rigorous approach with his implementation of convergence tests and epsilonic notation. Which again inspired Weier-strass and his followers to a consistent use of the epsilonic notation, which facilitated a distinction between first uniform and pointwize convergence and then a distinction between first uniform and pointwize continuity. By this there arose a need for more symbols and other forms of notation. Thus at the end of the 19 th century both the notation and the concepts of calculus were similar to modern calculus

Vejleder: Jesper Lützen
Censor: Poul Hjorth, DTU