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.
Author(s): Jesper Lützen
Edition: 1
Publisher: Oxford University Press
Year: 2023
Language: English
Pages: 304
City: New York
Tags: Classical Problems; Diorisms; Cube Duplication; Angle Trisection; Circle Quadrature; Complex Numbers; Euler; Quintic Insolvability; Impossible Integrals; Parallel Postulate; Hilbert; Gödel; Fermat's Last Theorem
Cover
Half Title Page
Title Page
Copyright Page
Preface
Contents
1. Introduction
1.1 The organization of the book
1.2 What is an impossibility theorem?
1.3 Meta statements and mathematical results
1.4 Why are impossibility results often misunderstood among amateur mathematicians?
1.5 Impossibility results in mathematics and elsewhere
1.6 A classification of mathematical impossibility results
1.7 Impossibility as a creative force
2. Prehistory: Recorded and Non-recorded Impossibilities
3. The First Impossibility Proof: Incommensurability
3.1 The discovery
3.2 The consequences of the impossibility theorem
3.3 Incommensurable quantities in Euclid’s Elements
4. Classical Problems of Antiquity: Constructions and Positive Theorems
4.1 Squaring a circle
4.2 Doubling the cube
4.3 Trisecting the angle
5. The Classical Problems: The Impossibility Question in Antiquity
5.1 Existence and constructability
5.2 Pappus on the classification of geometric problems
5.3 The quadrature of a circle
5.4 Using non-constructible quantities: Archimedes and Ptolemy
6. Diorisms: Conclusions about the Greeks and Medieval Arabs
6.1 Diorisms
6.2 Conclusion on impossibilities in Greek mathematics
6.3 Medieval Arabic contributions
7. Cube Duplication and Angle Trisection in the Seventeenth and Eighteenth Centuries
7.1 The seventeenth century
7.2 Descartes’s analytic geometry
7.3 Descartes on the duplication of a cube and the trisection of an angle
7.4 Descartes’s contributions
7.5 The eighteenth century
7.6 Montucla and Condorcet compared with Descartes
8. Circle Quadrature in the Seventeenth Century
8.1 “Solutions” and positive results
8.2 Descartes on the quadrature of a circle
8.3 Wallis on the impossibility of an analytic quadrature of a circle
8.4 Different quadratures of a circle
8.5 Gregory on impossibility proofs and the new analysis
8.6 Gregory’s argument for the impossibility of the algebraic indefinite circle quadrature
8.7 Huygens’ and Wallis’ critique of Gregory
8.8 Leibniz on the impossibility of the indefinite circle quadrature
8.9 Newton’s argument for the impossibility of the algebraic indefinite oval quadrature
8.10 Why prove impossibility
9. Circle Quadrature in the Eighteenth Century
9.1 Joseph Saurin (1659–1737)
9.2 Anonymous
9.3 Thomas Fantet De Lagny (1660–1734)
9.4 The enlightened opinion
9.5 D’ Alembert
9.6 The French Academy of Sciences. Condorcet
9.7 Enlightening the amateurs
9.8 Lambert and the irrationality of π
10. Impossible Equations Made Possible: The Complex Numbers
10.1 The extension of the number system: Wallis’s account
10.2 Cardano’s sophisticated and useless numbers
10.3 The unreasonable usefulness of the complex numbers
10.4 A digression about infinitesimals
11. Euler and the Bridges of Königsberg
12. The Insolvability of the Quintic by Radicals
12.1 Early results
12.2 Paolo Ruffini
12.3 Niels Henrik Abel
13. Constructions with Ruler and Compass: The Final Impossibility Proofs
13.1 Gauss on regular polygons
13.2 Wantzel
13.3 The quadrature of a circle
14. Impossible Integrals
14.1 Early considerations
14.2 Abel’s mostly unpublished results
14.3 Joseph Liouville on integration in algebraic terms
14.4 Liouville on integration in finite terms
14.5 Liouville on solution of differential equations by quadrature
14.6 Later developments
14.7 Concluding remarks on the situation c.1830
15. Impossibility of Proving the Parallel Postulate
15.1 The axiomatic deductive method
15.2 The parallel postulate and the attempts to prove it
15.3 Indirect proofs: Implicit non-Euclidean geometry
15.4 Non-Euclidean geometry: The invention
15.5 The help from differential geometry of surfaces
15.6. Conclusions
16. Hilbert and Impossible Problems
16.1 Impossibility as a solution; rejection of ignorabimus
16.2 Hilbert’s third problem: Equidecomposability
16.3 Hilbert’s seventh problem
16.4 Hilbert’s first problem
17. Hilbert and Gödel on Axiomatization and Incompleteness
17.1 The axiomatization of mathematics
17.2 Hilbert’s second Paris problem
17.3 The foundational crisis
17.4 Gödel’s incompleteness theorems
17.5 Hilbert’s tenth Paris problem
17.6 Conclusion
18. Fermat’s Last Theorem
18.1 Fermat’s contribution
18.2 Nineteenth-century contributions
18.3 The twentieth-century proof
19. Impossibility in Physics
19.1 The impossibility of perpetual motion machines
19.2 Twentieth-century impossibilities in physics
20. Arrow’s Impossibility Theorem
20.1 The theory of voting
20.2 Welfare economics
20.3 The Impossibility theorem
20.4 The Gibbard–Satterthwaite theorem
21. Conclusion
21.1 From unimportant non-results to remarkable “solutions”
21.2 From meta-statements to mathematical theorems
21.3 Different types of problems and different types of proofs
21.4 Pure and applied impossibility theorems
21.5 Controversies
21.6 Impossibility as a creative force
Recommended Supplementary Reading
References
Index
