What's the greatest discovery in the history of thought? Of course, it's a silly question - but it won't stop me from suggesting an answer. It's Plato's discovery of abstract objects. Most scientists, and indeed most philosophers, would scoff at this. Philosophers admire Plato as one of the greats, but think of his doctrine of the heavenly forms as belonging in a museum. Mathematicians, on the other hand, are at least slightly sympathetic. Working day-in and day-out with primes, polynomials and principle fibre bundles, they have come to think of these entities as having a life of their own. Could this be only a visceral reaction to an illusion? Perhaps, but I doubt it. The case for Platonism, however, needs to made carefully. Let's begin with a glance at the past.