A History of Mathematical Impossibility

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Beschreibung

Many of the most famous results in mathematics are impossibility theorems stating that something cannot be done. Good examples include the quadrature of the circle by ruler and compass, the solution of the quintic equation by radicals, Fermat's last theorem, and the impossibility of proving the parallel postulate from the other axioms of Euclidean geometry. This book tells the history of these and many other impossibility theorems starting with the ancient Greek proof of the incommensurability of the side and the diagonal in a square.

Lützen argues that the role of impossibility results have changed over time. At first, they were considered rather unimportant meta-statements concerning mathematics but gradually they obtained the role of important proper mathematical results that can and should be proved. While mathematical impossibility proofs are more rigorous than impossibility arguments in other areas of life, mathematicians have employed great ingenuity to circumvent impossibilities by changing the rules of the game. For example, complex numbers were invented in order to make impossible equations solvable. In this way, impossibilities have been a strong creative force in the development of mathematics, mathematical physics, and social science.

Über den Autor

Jesper Lützen is a historian of mathematics and the physical sciences. He is Professor Emeritus at the Department of Mathematical Sciences at the University of Copenhagen, where he has taught since 1989.

Inhaltsverzeichnis

1: Introduction

2: Prehistory: Recorded and Non-Recorded Impossibilities

3: The First Impossibility Proof: Incommensurability

4: The Classical Problems in Antiquity: Constructions and Positive Theorems

5: The Classical Problems: The Impossibility Question

6: Diorisms and Conclusions about the Greeks and the Medieval Arabs

7: Cube Duplication and Angle Trisection in the 17th and 18th Centuries

8: Circle Quadrature in the 17th Century

9: Circle Quadrature in the 18th Century

10: Impossible Equations Made Possible: The Complex Numbers

11: Euler and the Bridges of Königsberg

12: The Insolvability of the Quintic by Radicals

13: Constructions with Ruler and Compass: The Final Impossibility Proofs

14: Impossible Integrals

15: Impossibility of Proving the Parallel Postulate

16: Hilbert and Impossible Problems

17: Hilbert and Gödel on Axiomatization and Incompleteness

18: Fermat's Last Theorem

19: Impossibility in Physics

20: Arrow's Impossibility Theorem

21: Conclusion

Details

Erscheinungsjahr:	2023
Fachbereich:	Geometrie
Genre:	Importe , Mathematik
Rubrik:	Naturwissenschaften & Technik
Medium:	Buch
Inhalt:	Gebunden
ISBN-13:	9780192867391
ISBN-10:	0192867393
Sprache:	Englisch
Einband:	Gebunden
Autor:	Lützen, Jesper
Hersteller:	Oxford University Press
Verantwortliche Person für die EU:	Deutsche Bibelgesellschaft, Postfach:81 03 40, D-70567 Stuttgart, vertrieb@dbg.de
Maße:	236 x 157 x 25 mm
Von/Mit:	Jesper Lützen
Erscheinungsdatum:	26.04.2023
Gewicht:	0,658 kg