From Mathematical Practice to Minimal Mathematical Realism

Mathematical practice exhibits a pervasive lack of freedom: once relevant assumptions and rules are fixed, practitioners do not appear free to choose what follows. In this paper, I develop Minimal Mathematical Realism (MMR), according to which an Autonomous Mathematical Reality (AMR) underlies this phenomenon, while making no initial assumption about its intrinsic character or organisation. At this minimal level, the usual distinction between realist and non-realist accounts becomes less clear, since approaches rejecting familiar realist ontologies may nevertheless be compatible with MMR. The inference to an AMR is abductive: it provides a straightforward explanation of the stable mathematical determinations encountered across practitioners, representations, and domains. The differentiated character of these determinations also provides grounds for tentatively characterising the AMR, without assuming that familiar mathematical categories reproduce its intrinsic organisation. More speculatively, MMR raises the possibility that mathematical reality itself is non-optional: its non-existence may simply not be an alternative.