Modal structuralism and the problem of integration

Philosophy "Mathematical objects, structures and proofs" (thematic issue)

Abstract

Modal structuralism attempts to solve the problems of Platonism in the philosophy of mathematics. First, the paper presents the view that modal structuralism emerges out of Benacerraf’s arguments against Platonism and set-theoretic reductionist realism. Putnam’s account is showed to be another source of influence on modal structuralism. Second, the basic ideas of modal structuralism are reviewed, with special attention paid to how the translation of mathematical statements into modal sentences helps to avoid the set-theoretic grounding of such statements. However, since possible worlds are conceived as set-theoretic entities, the translation itself faces the problem of potential circular explanation. To solve the problem, Hellman suggests taking modalities as primitives, but his solution faces additional issues. Two of them are an unclear metaphysical status of the possible structures mathematicians make statements about and an obscure epistemic access mathematicians have to these structures. In order to avoid these issues, the paper suggests combining modal structuralism with modal normativism. According to the latter, modal statements are not about objects or facts but about linguistic rules. Since modal normativists interpret the possible structures as non-metaphysical entities, the problem of epistemic access to such structures transforms into the problem of the agent’s knowledge of the semantic rules of mathematical language. It is also pointed out that modal normativism might solve another set of structuralist problems, not specifically concerned with modalities, e.g., the problem of objects’ dependence on the structures in which they are embedded.