A scholastic-realist modal-structuralism

Pietarinen

pp. 127-138

in: Peter Schroeder-Heister (ed), Logic and Philosophy of Science in Nancy (I), Philosophia Scientiae 18 (3), 2014.

Abstract

How are we to understand the talk about properties of structures the existence of which is conditional upon the assumption of the reality of those structures? Mathematics is not about abstract objects, yet unlike fictionalism, modal-structuralism respects the truth of theorems and proofs. But it is nominalistic with respect to possibilia. The problem is that, for fear of reducing possibilia to actualities, the second-order modal logic that claims to axiomatise modal existence has no real semantics. There is no cross-identification of higher-order mathematical entities and thus we cannot know what those entities are. I suggest that a scholastic notion of realism, interspersed with cross-identification of higher-order entities, can deliver the semantics without collapse. This semantics of modalities is related to Peirce's logic and his pragmaticist philosophy of mathematics.

Cited authors

Peirce Charles Sanders