This book presents a historical account of Felix Klein's "Comparative Reflections on Recent Research in Geometry" (1872), better known as his "Erlangen Program.” Originally conceived and written when Klein was collaborating with Sophus Lie, this bold essay initially made little impression on contemporary researchers. Decades later, however, it eventually became a famous classic. Eminent mathematicians hailed Klein’s main message – the role of invariants of transformation groups in geometry – as presaging major developments in mathematics and physics.
The first part of this book focuses on the prehistory surrounding Klein’s “Erlangen Program,” stressing the motivations that led Klein to write it. The core of the book (Part II) then presents a new translation of Klein's original text, followed by detailed textual analysis aimed at guiding the reader through its rather terse and opaque prose. Part III deals with its complicated reception history, treated in four periods spanning the years from 1872 to 1930. This culminated during Klein’s lifetime with his efforts to promote the "Erlangen Program” as a framework for interpreting Einstein’s theory of relativity. After his death in 1925, the viability of this framework became a contentious issue among leading differential geometers. Part IV looks back on the transformations in mathematics that led to a modernized interpretation of Klein’s message. The book also explores in depth how the growing fame of the “Erlangen Program” undermined Klein’s friendship with Sophus Lie, leading to a dramatic public break between them in 1893.
Beyond the "Erlangen Program” itself, this book deals with many of Felix Klein’s other works. As an introduction to a largely forgotten world of ideas, this study will appeal not only to experts but also to graduate students and all those with a serious interest in the history of modern mathematics.
Preface.- Introduction.- Part I. Prehistory of the Erlangen Program.-
1 Klein as a Young Geometer.-
2. Klein Encounters Sophus Lie.-
3. Klein on
Cayleys Projective Metric.- Part II. Kleins Erlangen Program with
Commentary.-
4. Kleins Erlangen Program.-
5. Textual Analysis of Kleins
Erlangen Program.- Part III. Four Phases of Reception and Transformation.-
6. First Phase of Reception, 18731889.-
7. Second Phase of Reception,
18901899.-
8. Third Phase of Reception, 19001916.-
9. Fourth Phase of
Reception, 19171930.- Part IV. Reconsiderations.- 10 Historical
Reflections.- Bibliography.- Name Index.
David E. Rowe is professor emeritus for history of mathematics and natural sciences at Mainz University. Much of his research has centered on mathematics in Germany from 1850 to 1933 with special focus on the Göttingen mathematical tradition. He is the author and editor of several books and numerous articles and reviews, many published by Springer in both English and German. Among his most recent publications are Emmy Noether Mathematician Extraordinaire, Springer, 2021, Jenseits von Flachland: Mathematische Grenzüberschreitungen und ihre Auswirkungen, co-authored with Klaus Volkert, Springer Spektrum, 2023, and the translation of the biography Felix Hausdorff: Mathematician, Philosopher, Man of Letters, by Egbert Brieskorn and Walter Purkert, Birkhäuser, 2024.
