Conflicts Between Generalization, Rigor, and Intuition - (Sources and Studies in the History of Mathematics and Physic) by Gert Schubring

Book Synopsis

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 1811--precipitating the change back to the synthetic method and replacing the limit method by the method of the quantités infiniment petites--significantly altered the teaching of analysis at this first modern institution of higher education, an institution originally founded as a citadel of the analytic method.

From the Back Cover

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 18^th 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.

Review Quotes

From the reviews:

"This is a very ambitious book, both in its methodology and in the amount of material it addresses. ... The topic is of major interest, focusing on two concepts that extended the idea of what was meant by 'number' ... . this book remains a major contribution. The rich and detailed account of textbooks and educational institutions, and the key passages and events Schubring highlights ... add greatly to our understanding of the history of mathematics in one of its most exciting periods." (Judith V. Grabiner, SIAM Review, Vol. 48 (2), 2006)

"The present book is a voluminous and detailed study of the conceptual developments of negative numbers and infinitesimals from the prehistory of the calculus to the end of the nineteenth century. ... It stands out as special by treating many primary mathematical sources that are rarely subjected to historical study ... . this volume presents an important new contextualised perspective on the history of negative numbers and infinitesimals. It includes a rich variety of institutional and philosophical discussions ... ." (Henrik Kragh Sørenson, Zentralblatt MATH, Vol. 1086, 2006)

"This deep and important epistemological study analyses the evolution of concepts fundamental to mathematical analysis up to the nineteenth century. ... The author examines how concepts were generalized and differentiated and pays particular attention to the role of symbolism. He critically reviews the work of other authors who have treated the same historical periods." (E. J. Barbeau, Mathematical Reviews, Issue 2006 d)