EXCERPT In this second part, we present brief case studies from the landscape of contemporary mathematics (1950–2000). Our strategy will consist in providing mathematical information (primarily conceptual and, to a lesser extent, technical information) that is usually taken to fall outside the scope of philosophy – information whose philosophical distillation and discussion will occupy us in part 3. Nevertheless, although the primary objective of this second part is to expand the concrete mathematical culture of the reader, we will also go on to indicate and briefly discuss a few generic lines of tension, both methodological and creative, that a complete philosophical understanding of mathematics will have to confront…