Conflicts Between Generalization, Rigor, and Intuition

• 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.

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 Sorenson, 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)