Should we accept the existence of numbers as abstract objects?

Abstract

This paper argues that we should accept the existence of numbers as abstract objects. I begin by looking at the Indispensability Argument as a positive argument for the existence of mathematical objects. I look at various discussions that challenge the Indispensability Argument, most notably that we can dismiss the doctrine of Confirmational Holism by looking to an argument that we use figurative talk in science, such as idealisations. This is the argument that, like idealisations, mathematical objects are only used in scientific theories figuratively and as such their existence is not confirmed by the success of the scientific theories they are included in. In order for this objection to stand, it needs to explain what figurative uses are and so I look at Mathematical Fictionalism, which aims to explain these uses for mathematical statements. I look at Yablo’s fictionalism, as the strongest argument for explaining this, and explain how he hopes to dismiss our commitment to the existence of mathematical objects by arguing that they are ‘creatures of existential metaphor’. Finally, I raise issues with this explanation and conclude that, without further argument, the fictionalist position does not convincingly dismiss our commitment to the existence of mathematical objects and so we should accept the existence of numbers as abstract objects.