Understanding and Mathematical Fictionalism

In a recent paper in this journal, Mark Balaguer develops and defends a new version of mathematical fictionalism, what he calls 'Hermeneutic non-assertivism', and responds to some recent objections to mathematical fictionalism that were launched by John Burgess and others. In this paper I provide some fairly compelling reasons for rejecting Hermeneutic non-assertivism - ones that highlight an important feature of what understanding mathematics involves (or, as we shall see, does not involve).