Thomas Kuhn made his reputation by pointing out that science is not just a steady series of rational developments, but is subject to paradigm shifts – revolutions in which the entire point of view changes. A less flattering word for the process is ‘fashion’. But what is fashion other than a paradigm shift in a rather trivial area of human activity? It is often thought that mathematics is immune to paradigm shifts. After all, mathematics consists of perfect truths, proven by impeccable logic. Doesn't it? Well, no, not really – but even if it did, that wouldn't prevent paradigm shifts. Paradigm shifts are not about the internal logic; they are about the external viewpoint. They are about seeing the same facts in a new light, incorporating them into a new framework.