How to Refer to Abstract Objects
J. Davies · TSpace (University of Toronto) · 2016
Consider the statement “7 + 5 = 12”. It appears to be true. Moreover, it appears to entail that something is identical to 7 + 5. But this thing that is identical to 7 + 5 does not appear to be located in space or time – where and when would we find it? Nor does it appear to enter into any causal interactions – what causes can influence it, and what events has it produced? Mathematical platonism is the view that all these intuitive appearances are correct: many of our mathematical statements are true statements about non-spatiotemporal and non-causal objects, or as philosophers say, abstract objects. Mathematical platonism stands in direct tension with many contemporary theories about how our thoughts and utterances can be about mind-independent objects, as most of those theories explain how we can think and speak about such objects in causal terms. This observation gives rise to the following argument, which I call the 'semantic argument against mathematical platonism'. For our any one of our expressions to refer to an object o, it must be possible for o to stand in at least one causal relation. It is impossible for abstract mathematical objects to stand in causal relations. Therefore reference to abstract mathematical objects is impossible. But if reference to abstract mathematical objects is impossible, then hardly any of our mathematical statements are true – at least when read at face value. This substantially undermines mathematical platonism. This dissertation defends mathematical platonism against the semantic argument. It has four chapters and a brief conclusion. I ultimately describe a package of views that may allow the platonist to resist the semantic argument. In particular this package of views allows the platonist to claim that not only can we refer to abstract mathematical objects, but we can refer singularly to them. Hence not only are apparent mathematical truths like “7 + 5 = 12” indeed true, but they say what they appear to say.