Mathematics is almost always an insider's affair. But sometimes things happen within the mathematical community that have a relevance, and perhaps also an interest, beyond the tribe itself. The Grundlagenstreit of the 1920s is such an example. In this review essay we tell this story with focus on the main actors involved, David Hilbert in Göttingen and L. E. J. Brouwer in Amsterdam. We shall see how fine points concerning the existence of mathematical objects, the question of the editorship of the Mathematische Annalen, and the attempts to resume normal scientific contacts between French and Germans scientists after the First World War led to an unusual bitter conflict within the tribe and beyond. But even if the effects of the fight were at the time negative, the long range outcome was positive. Hilbert's work on the foundation of mathematics is still a powerful influence on current research, and Brouwer's view on the constructive foundation of mathematics, which at the time inspired both Husserl and Wittgenstein, is today of increasing importance in the evolving science of logic and computing.