Modal structuralism simplified

Abstract

Since Benacerraf’s ‘What Numbers Could Not Be, ’ there has been a growing interest in mathematical structuralism. An influential form of mathematical structuralism, modal structuralism, uses logical possibility and second order logic to provide paraphrases of mathematical statements which don’t quantify over mathematical objects. These modal structuralist paraphrases are a useful tool for nominalists and realists alike. But their use of second order logic and quantification into the logical possibility operator raises concerns. In this paper, I show that the work of both these elements can be done by a single natural generalization of the logical possibility operator.

Nominalism and Mathematical Objectivity

“…(ii) How about dispensing with second-order logic by making the relations and properties in (M) fixed? For instance, let us follow Berry (2018) and paraphrase ϕ as 27 :…”

“…PA ♢ comes by combing four first-order axioms of PA with □ ,s P(0) ∧ ( ∀x)( ∀y)(P(x) ∧ s(x, y) → P(y) → (∀x)( (x) → P(x)). 28 It is worth mentioning that Berry (2018) motivates her simplified modal structuralism as an alternative to Hellman's canonical modalism in order to solve the quantifying in problem and the incompossible problem. So there is no problem for her to take some mathematical facts for granted, but this is not available to a nominalist.…”

David Armstrong on the Metaphysics of Mathematics

“…Some restriction of the usual comprehension axiom will be needed to accommodate Armstrong's contention that there are no uninstantiated properties. For a version of modal structuralism that does not require second order quantification, see Berry (2018). The parenthetical "and perhaps more than sufficient" is there to indicate that Armstrong's was an inexact conception of truthmaking, to use Kit Fine's terminology-see (2017).…”