Application and ontology in mathematics: a defence of fictionalism

The aim of this thesis is to defend fictionalism as a response to the mathematical placement problem. As we will see, against the backdrop of philosophical naturalism, it is difficult to see how to fit mathematical objects into our best total scientific theory. On the other hand, the indispensability argument seems to suggest that science itself mandates ontological commitment to mathematical entities. My goal is to undermine the indispensability argument by presenting an account of applied mathematics as a kind of revolutionary prop-oriented make-believe, the content of which is given by a mapping account of mathematical applications.
This kind of fictionalism faces a number of challenges from various quarters. To begin with, we will have to face the challenge of a different kind of indispensability argument, one that draws ontological conclusions from the role of mathematical objects in scientific explanations. We will then examine one recent theory of mathematical scientific representation, and discover that the resulting position is Platonistic. At this point we will introduce our fictionalist account, and see that it defuses the Platonist consequences of mathematical representation.
The closing chapters of the thesis then take a metaphilosophical turn. The legitmacy of a fictionalist response to the mathematical placement problem is open to challenge from a metaphilosophical perspective in two different ways: on the one hand, some modern pragmatists have argued that this kind of metaphysics relies on questionable assumptions about how langauge works. On the other, some modern philosophers have developed forms of metaontological anti-realism that they believe undermine the legitimacy of philosophical work in metaphysics. In the final two chapters I defend the fictionalist account developed here against these sceptical claims.
I conclude that the fictionalist account of applied mathematics offered here is our best hope for coping with the mathematical placement problem.