The previous two sections critically discussed a modal version of eliminative structuralism (though to my mind, the objections raised weren’t particularly telling). Parsons now moves on characterize his own preferred “noneliminative structuralism”, and responds to some potential obections.
I wish I could give a sharp characterization of the position Parsons wants to occupy here in the longest section of his book. But I do have to confess bafflement. “We have emphasized the point going back to Bernays that reference to mathematical objects is relative to a background structure.” Further, structures aren’t themselves objects, and “[Parsons’s] structuralist account of a particular kind of mathematical object does not view statements about that kind of object as about structures at all”. But surely there’s thus far nothing that e.g. the Fregean need dissent from. The Fregean can agree that numbers, for example, don’t come (so to speak) independently, but come all together forming an intrinsically order structured: and in identifying the number 42 as such, we necessarily give its position in relation to other numbers. So what more is Parsons saying about (say) numbers that distinguishes his position? Well, I’ve read the section three times and I’m still rather lost, and won’t ramble here. If any other reader of the book can offer some crisp clarifying comments, I for one would be very grateful!