link.springer.com
How the laws of logic lie about mathematical objects - Synthese
In ontological debates on the existence of mathematical objects, it has long been taken for granted that if we take our mathematical discourse at face value, it follows from the fact that our true mathematical statements refer to mathematical objects that mathematical objects exists; reference in true statements entails existence. In this paper, I argue that there are positions available in the philosophy of logic that allow us to dislodge this assumption, allowing for a nominalist position according to which statements such as ‘there are four prime numbers between 1 and 10’ are true and genuinely refer to numbers, without a corresponding statement asserting the existence of numbers following from it. Consequently, there is an overlooked nominalist position according to which objects just are what singular terms refer to and that reference is successful when those singular terms figure in true statements, with corresponding existence statements being literally false. I argue that this view is not only coherent, but that it does not entail that there are objects that do not exist.
