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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.
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
Description
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.
