A History of Mathematical Impossibility

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Jesper Lützen (Department of Mathematical Sciences University of Copenhagen, Department of Mathematical Sciences University of Copenhagen)

A History of Mathematical Impossibility

Jesper Lützen (Department of Mathematical Sciences University of Copenhagen, Department of Mathematical Sciences University of Copenhagen)

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English

Oxford University Press
26 January 2023

Mathematical foundations; Calculus & mathematical analysis; History of mathematics

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.

By: Jesper Lützen (Department of Mathematical Sciences University of Copenhagen Department of Mathematical Sciences University of Copenhagen)
Imprint: Oxford University Press
Country of Publication: United Kingdom
Dimensions: Height: 240mm, Width: 163mm, Spine: 20mm
Weight: 1g
ISBN: 9780192867391
ISBN 10: 0192867393
Pages: 304
Publication Date: 26 January 2023
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

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

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.

Reviews for A History of Mathematical Impossibility

This book is intended as a semi-popular volume: in it, the author eschews mathematical or historical technicalities, instead providing succinct yet rich accounts that neatly convey the main conceptual innovations and transformations at the heart of the episodes discussed therein...The writing is clear and engaging. * Historia Mathematica *