A History of Mathematical Impossibility

Impossibility theorems, such as the quadrature of the circle and Fermat's last theorem, have historically been considered unimportant meta-statements, but they have gradually gained importance as proper mathematical results. Mathematicians have employed ingenuity to circumvent impossibilities by changing the rules of the game, such as inventing complex numbers to make impossible equations solvable.

Format: Hardback
Length: 304 pages
Publication date: 26 January 2023
Publisher: Oxford University Press

Many of the most renowned results in mathematics are impossibility theorems that assert that something is impossible to achieve. Excellent 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 chronicles the history of these and many other impossibility theorems, beginning 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 has evolved over time. Initially, they were regarded as rather unimportant meta-statements concerning mathematics, but gradually they assumed the role of significant 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 to make impossible equations solvable. In this way, impossibilities have been a powerful creative force in the development of mathematics, mathematical physics, and social science.

Weight: 658g
Dimension: 163 x 242 x 24 (mm)
ISBN-13: 9780192867391