Conflicts Between Generalization, Rigor, and Intuition

Conflicts Between Generalization, Rigor, and Intuition undertakes a historical analysis of the development of two mathematical concepts -negative numbers and infinitely small quantities, mainly in France and Germany, but also in Britain, and the different paths taken there.

This book not only discusses the history of the two concepts, but it also introduces a wealth of new knowledge and insights regarding their interrelation as necessary foundations for the emergence of the 19th century concept of analysis. The historical investigation unravels several processes underlying and motivating conceptual change: generalization (in particular, algebraization as an agent for generalizing) and a continued effort of intuitive accessibility which often conflicted with likewise desired rigor. The study focuses on the 18th and the 19th centuries, with a detailed analysis of Lazare Carnot's and A. L. Cauchy's foundational ideas.

By researching the development of the concept of negative and infinitely small numbers, the book provides a productive unity to a large number of historical sources. This approach permits a nuanced analysis of the meaning of mathematical ideas as conceived of by 18th and 19th century scientists, while illustrating the authors' actions within the context of their respective cultural and scientific communities. The result is a highly readable study of conceptual history and a new model for the cultural history of mathematics.

Zusammenfassung
This volume is, as may be readily apparent, the fruit of many years' labor in archives and libraries, unearthing rare books, researching Nachlässe, and above all, systematic comparative analysis of fecund sources. The work not only demanded much time in preparation, but was also interrupted by other duties, such as time spent as a guest professor at universities abroad, which of course provided welcome opportunities to present and discuss the work, and in particular, the organizing of the 1994 International Graßmann Conference and the subsequent editing of its proceedings. If it is not possible to be precise about the amount of time spent on this work, it is possible to be precise about the date of its inception. In 1984, during research in the archive of the École polytechnique, my attention was drawn to the way in which the massive rupture that took place in 1811precipitating the change back to the synthetic method and replacing the limit method by the method of the quantités infiniment petitessignificantly altered the teaching of analysis at this first modern institution of higher education, an institution originally founded as a citadel of the analytic method.

Inhalt
Question and Method.- Paths Toward Algebraization Development to the Eighteenth Century. The Number Field.- Paths toward Algebraization The Field of Limits: The Development of Infinitely Small Quantities.- Culmination of Algebraization and Retour du Refoulé.- Le Retour du Refoulé: From the Perspective of Mathematical Concepts.- Cauchy's Compromise Concept.- Development of Pure Mathematics in Prussia/Germany.- Conflicts Between Confinement to Geometry and Algebraization in France.- Summary and Outlook.