In sum, then, we might put things like this. Parsons has defended an ‘internalist’ argument — an argument from “within mathematics” — for the uniqueness of the numbers we are talking about in our arithmetic, whilst arguing against the need for (or perhaps indeed, the possibility of) an ‘externalist’ justification for our intuition of uniqueness.

Can we rest content with that? Some philosophers would say we can get more — and Parsons briefly discusses two, Hartry Field and Shaughan Lavine, though he gives fairly short shrift to both. Field has argued that we can appeal to a ‘cosmological hypothesis’ together with an assumption of the determinateness of our physical vocabulary to rule out non-standard models of our applicable arithmetic. Parsons reasonably enough worries: “If our powers of mathematical concept formation are not sufficient [to rule out nonstandard models], then why should our powers of physical concept formation do any better?” Lavine supposes that our arithmetic can be regimented as a “full schematic theory” which is in fact stronger than the sort of theory with open-ended induction that we’ve been considering, and for which a categoricity theorem can be proved. But Parsons finds some difficulty in locating a clear conception of exactly what counts as a full schematic theory — a difficulty on which, indeed, I’ve commented elsewhere on this blog.

In both cases, I think Parsons’s points are well taken: but his discussions of Field and Lavine are brief, and more probably needs to be said (though not here).