Hazhir Roshangar

Taking mathematics as a language based on empirical experience, I argue for an account of mathematics in which its objects are abstracta that describe and communicate the structure of reality based on some of our ancestral interactions with their environment. I argue that mathematics as a language is mostly invented. Nonetheless, in being a general description of reality it cannot be said that it is fictional; and as an intersubjective reality, mathematical objects can exist independent of any one person’s mind.

I argue that, within a bivalent classical framework, absolute nothingness cannot be the target of proper reference if such reference requires a determinate relatum. Here, absolute nothingness is understood through a meta-theoretical limiting specification according to which no object and no distinction obtain. The problem revolves around finding a way for this specification to remain available to thought and discourse without reifying its intended target. To address this issue, I distinguish between specification, attempted reference, and proper reference to show how a specification can govern meaningful inference without thereby securing a determinate relatum. To register the failure of proper reference, I introduce the concept of an exclusion surrogate that records universal non-satisfaction within a theory without supplying the intended referent. This provides a classical alternative to Priest’s dialetheic treatment; that is, under these assumptions, the availability/non-reification tension does not require a contradictory referent that both is and is not an object.