Mathematics’ Modal Content

Otávio Bueno (University of Miami)

On the surface, mathematical discourse and mathematical objects and structures may be thought as not being modal at all. Mathematical language is purely extensional, and the corresponding objects and structures, on the usual interpretation, are all abstract. How could mathematics possibly have any modal content?

In this paper, I argue that appearances are deceptive, and that modality is integral to the very content of mathematics. Modality emerges in different ways. First, as opposed to the standard model-theory approach, the very concept of logical consequence, crucial to mathematical practice, is modal in nature: in a valid argument, the conjunction of the premises and the negation of the conclusion is not possible. Second, theorems, being deductively derived from principles and assumptions regarding the relevant mathematical objects and structures, encode possibilities: what can or cannot obtain, given the principles, assumptions, and the underlying logic. Third, the fact that the mathematical content of a result changes substantially with a change in logic, as is clear when classical and constructive mathematics are compared, is the reflection of a modal variation: what is possible or not, given the logic under consideration. Finally, the application of mathematics crucially relies on the modal content of mathematics, highlighting the possibilities or impossibilities of certain empirical situations, properly interpreted. I will examine and defend each of these points, thus highlighting mathematics’ integral modal content.