Neo-Fregeanism Reconsidered
Mathematical Platonism is the view that mathematical objects exist. Traditional Platonists believe that a world with no mathematical objects is consistent; subtle Platonists believe that such a world would be inconsistent. The easiest way of getting a handle on traditional Platonism is by imagining a creation myth. On the first day God created light; by the sixth day, she had created a large and complex world, including black holes, planets and sea-slugs. But there was something left to be done. So on the seventh day she created mathematical objects. Only then did she rest. On this view, it is easy to make sense of a world with no mathematical objects: it is just like the world we are considering, except that God rested on the seventh day. The crucial feature of this creation myth is that God needed to do something extra in order to bring about the existence of mathematical objects: something that wasn’t already in place when she created black holes, planets and sea-slugs. According to subtle Platonists, this is a mistake. A subtle Platonist believes that for the number of the Fs to be eight just is for there to be eight planets. So when God created eight planets she thereby made it the case that the number of the planets was eight. More generally, subtle ∗For their many helpful comments I am grateful to Roy Cook and Matti Eklund, and to audiences at the University of Connecticut and the University of St Andrews.