Impossibility of a Solution to Certain Problems Plays an Important Role

In modern mathematics, the question of the impossibility of a solution to certain problems plays an important role, and attempts to answer such questions have often been the occasion of discovering new and fruitful fields for research.

We recall in this connection the demonstration by Abel of the impossibility of solving an equation of the fifth degree by means of radicals, the discovery of the impossibility of demonstrating the axiom of parallels, and finally the theorems of Hermite and Lindeman concerning the impossibility of constructing by algebraic means the numbers e and π.

--From (Foundations of Geometry)