Plato's heaven

Abstract

The laws of mathematics are not merely human inventions or creations. They simply ‘are’; they exist quite independently of the human intellect . M. C. Escher (1898–1972) Mathematical realism or Platonism is the philosophical position that mathematical statements such as ‘there are infinitely many prime numbers’ are true and that these statements are true by virtue of the existence of mathematical objects – prime numbers, in this case. That all seems fine until you think about the nature of the objects being posited. Where are these numbers? What are they like? How do we know about them? What about all the other mathematical objects: sets, functions, Hilbert spaces, and the like? Do all these exist as well? Are all mathematical objects made up of the same basic ingredients – sets, perhaps – or are they each a distinct kind of thing? Are these mathematical objects abstract or do they have causal powers and space-time locations? In any case, what is their relationship to the physical world? And most difficult of all: if mathematical knowledge is knowledge of these mathematical entities how do we come by such knowledge? Negotiating a set of answers to these questions, unsurprisingly, leads to a variety of different realist positions. In this chapter we will very briefly consider a few of the realist positions on offer, before looking in more detail at an influential argument for mathematical realism.