Sources and Studies in the History of Mathematics and Physical Sciences

Conflicts Between Generalization, Rigor, and Intuition

Number Concepts Underlying the Development of Analysis in 17th-19th Century France and Germany

Authors: Schubring, Gert

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About this book

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.

Reviews

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)


Table of contents (2 chapters)

  • Paths Toward Algebraization — Development to the Eighteenth Century. The Number Field

    Pages 15-150

  • Development of Pure Mathematics in Prussia/Germany

    Pages 481-565

Buy this book

eBook n/a
  • ISBN 978-0-387-28273-2
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
Hardcover n/a
  • ISBN 978-0-387-22836-5
  • Free shipping for individuals worldwide
Softcover n/a
  • ISBN 978-1-4419-1987-8
  • Free shipping for individuals worldwide
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Bibliographic Information

Bibliographic Information
Book Title
Conflicts Between Generalization, Rigor, and Intuition
Book Subtitle
Number Concepts Underlying the Development of Analysis in 17th-19th Century France and Germany
Authors
Series Title
Sources and Studies in the History of Mathematics and Physical Sciences
Copyright
2005
Publisher
Springer-Verlag New York
Copyright Holder
Springer-Verlag New York
eBook ISBN
978-0-387-28273-2
DOI
10.1007/0-387-28273-4
Hardcover ISBN
978-0-387-22836-5
Softcover ISBN
978-1-4419-1987-8
Series ISSN
2196-8810
Edition Number
1
Number of Pages
XIV, 678
Number of Illustrations and Tables
22 b/w illustrations
Topics