A History of Mathematical Impossibility

Abstract

Abstract Mathematical theorems stating that a problem cannot be solved using specific means are numerous. This book follows the history of such impossibility theorems from Greek antiquity through the early twentieth century. It reveals that many impossibility statements started out as meta-statements but ended up as mathematical theorems that were proved by mathematical methods. Until the nineteenth century, impossibility theorems were often considered of secondary interest compared with positive results. This changed during the nineteenth century and today impossibility results are among the most famous and popular theorems of mathematics. The book deals with some of the celebrated impossibility theorems in pure mathematics such as the quadrature of the circle, the duplication of the cube, the trisection of the angle, Fermat’s last theorem, the impossibility of proving the parallel postulate and Gödel’s theorem, as well as some theorems from applied mathematics such as Arrow’s impossibility theorem. Although an impossibility may sound as a negative result, impossibilities have in fact acted as a creative force in the history of mathematics, challenging mathematicians to circumvent the impossibility. The introduction of complex numbers is a case in point.