Duality in 19th and 20th Century Mathematical Thinking / edited by Ralf Krömer, Emmylou Haffner.

Format:

Book

Contributor:

Krömer, Ralf, editor.

Haffner, Emmylou, editor.

Series:

Science Networks. Historical Studies, 2296-6080 ; 63

Language:

English

Subjects (All):

Mathematics.

History.

Algebra, Homological.

Algebra, Universal.

History of Mathematical Sciences.

Category Theory, Homological Algebra.

General Algebraic Systems.

Local Subjects:

History of Mathematical Sciences.

Category Theory, Homological Algebra.

General Algebraic Systems.

Physical Description:

1 online resource (962 pages)

Edition:

1st ed. 2024.

Place of Publication:

Cham : Springer Nature Switzerland : Imprint: Birkhäuser, 2024.

Summary:

This volume brings together scholars across various domains of the history and philosophy of mathematics, investigating duality as a multi-faceted phenomenon. Encompassing both systematic analysis and historical examination, the book endeavors to elucidate the status, roles, and dynamics of duality within the realms of 19th and 20th-century mathematics. Eschewing a priori notions, the contributors embrace the diverse interpretations and manifestations of duality, thus presenting a nuanced and comprehensive perspective on this intricate subject. Spanning a broad spectrum of mathematical topics and historical periods, the book uses detailed case studies to investigate the different forms in which duality appeared and still appears in mathematics, to study their respective histories, and to analyze interactions between the different forms of duality. The chapters inquire into questions such as the contextual occurrences of duality in mathematics, the influence of chosen forms of representation, the impact of investigations of duality on mathematical practices, and the historical interconnections among various instances of duality. Together, they aim to answer a core question: Is there such a thing as duality in mathematics, or are there just several things called by the same name and similar in some respect? What emerges is that duality can be considered as a basic structure of mathematical thinking, thereby opening new horizons for the research on the history and the philosophy of mathematics and the reflection on mathematics in general. The volume will appeal not only to experts in the discipline but also to advanced students of mathematics, history, and philosophy intrigued by the complexities of this captivating subject matter.

Contents:

Preface

Chapter 1. Ralf Krömer, Emmylou Haffner: Introduction

Part I. The 19th century heritage

Chapter 2. Michael Friedman: On “contour apparent”, “courbe de contact” and ramification curves: Duality between a principle and a tool

Chapter 3. Juan Luis Gastaldi: De Morgan’s De Morgan’s Laws: Duality in the Emergence of Formal Logic

Chapter 4. Dirk Schlimm: The emergence of duality in 19th-century algebra of logic

Chapter 5. Emmylou Haffner: Duality as a guiding light in the genesis of Dedekind’s Dualgruppen

Chapter 6. Erhard Scholz: From Grassmann complements to Hodge-duality

Part II. From topology to groups

Chapter 7. Stephan Oltmanns, Ralf Krömer, Klaus Volkert: Duality theorems in topology

Chapter 8. Ralf Krömer: The historical development of Pontrjagin duality

Chapter 9. Harald Kümmerle: Tannaka Tadao’s 1938 paper on the duality of noncommutative topological groups and its historical background

Chapter 10. Christophe Eckes: Philosophical and mathematical duality in Albert Lautman’s work

Part III. Functional analysis and related fields

Chapter 11. Alessa Waldvogel: The development of dual spaces in functional analysis

Chapter 12. Frédéric Jaëck: Duality in Banach’s 1929 work on functionals

Chapter 13. Ralf Krömer: Marshall Stone and duality: from differential equations to Boolean algebras

Chapter 14. Ralf Krömer: Duality à la Bourbaki

Chapter 15. Jesper Lützen: Duality and Distributions: An Application of Topological Vector Spaces

Part IV. The post-war outlook

Chapter 16. Tinne Hoff Kjeldsen: From duality in mathematical programming to Fenchel duality and convex analysis: Duality as a force of inspiration in the creation of new mathematics

Chapter 17. Jean-Pierre Marquis: An Historical Perspective on Duality and Category Theory

List of abbreviations

Bibliography

Author index.

Notes:

Includes bibliographical references and index.

ISBN:

9783031597978

3031597974