Chapter 3 of Mathematical Thought and Its Objects is called “Modality and structuralism”. Before turning to discuss modal structuralism in Secs. 16 and 17, Parsons discusses what kind modality it might involve. Setting aside epistemic modalities as not to the present purpose, he considers (i) physical (or natural) necessity, (ii) metaphysical necessity (truth in all possible worlds), (iii) mathematical necessity, (iv) logical necessity (meant in a narrow sense that can be explicated model-theoretically).

Parsons argues that we don’t want to spell out a modal structuralism in terms of (i) natural modalities: “it demands too much to ask that the structures considered in mathematics be physically possible; indeed, in the case of higher set theory, there is every reason to believe that they are not physically possible.” I’ll buy that.

Second, Parsons argues that logical possibility — in the sense explicated via the idea of there being a suitable model — reveals itself as itself a mathematical notion, given that models are (at least typically) mathematical entities. So(?), “It is very doubtful that a generous notion of logical possibility would be distinguishable in a principled way from … mathematical possibility.”

But there is surely something rather odd here. For the idea, to repeat, is that we explicate “it is logically possible that P” (in the generous sense of allowed-at-least-by-considerations-of-logical-form, that runs beyond metaphysical possibility) in terms of there being a mathematical model on which P can be interpreted as true. It seems we don’t have a modality in the explanans here. Indeed, Parsons himself remarks on the common view that a mathematical truth (falsehood) is necessarily true (false): and on that view the very idea of a kind of “mathematical possibility” distinguished from plain truth evaporates.

So I’m left puzzled when Parsons concludes that the two runners for the kind of modality that might be involved in a modal structuralism are metaphysical modality and mathematical modality: for I just don’t have a grip on the latter notion.

(Relatedly, Parsons reads Putnam as holding that “it is mathematically possible that there should be no sets of uncountable rank, although it is a theorem of ZF that there are such sets”. Again, I really just don’t know how to construe that “mathematically possible” if that is supposed to be neither epistemic nor equivalent to “true”.)