Parsons’s Mathematical Objects: Sec. 54, Arithmetic

How does arithmetic fit into the sort of picture of the role of reason and so-called “rational intuition” drawn in Secs. 52 and 53?

The bald claim that some basic principles of arithmetic are “self-evident” is, Parsons thinks, decidedly unhelpful. Rather, “in mathematical thought and practice, the axioms of arithmetic are embedded in a rather dense network … [which] serves to buttress [their] evident character … so that in that respect their evident character does not just come from their intrinsic plausibility.” Moreover, there is a subtle interplay between general principles and elementary arithmetical claims — a dialectic “between attitudes towards mathematical axioms and rules and methodological or philosophical attitudes having to do with constructivity, predicativity, feasibility, and the like”. Which, as Parsons notes, is all beginning to sound rather Quinean. How is his position distinctive?

Not by making any more play with talk of “rational intuition”, which made its temporary appearance in Sec. 53 just as a way of talking about what is intrinsically plausible: indeed, the idea that the axioms of arithmetic derive a special status from being grounded in rational intuition is said to be “in an important way misleading”. Where Parsons does depart from Quine — and it is no surprise to be told, at this stage in the book! — is in holding that some elementary arithmetic principles can be intuitively known in the Hilbertian sense he discussed in earlier chapters. And the main point he seems to want to make in this chapter is that, as we move to more sophisticated areas of arithmetic which cannot directly be so grounded, so “the conceptual or rational element in arithmetical knowledge becomes much more prominent”, the web of arithmetic isn’t thereby totally severed from intuitive knowledge grounded in intuitions of stroke strings and the like. It is still the case that “an intuitive domain witnesses the possibility of the structure of numbers”.

Of course, how impressed we are by that claim will depend on how well we think Parsons defended his conception of intuitive knowledge in earlier chapters (and I’m not going to go over that ground again now, and nor indeed does Parsons). And what grounds the parts of arithmetic that don’t get rooted in Hilbertian intuition? To be sure, those more advanced parts can get tied to other bits of mathematics, notably set theory, so there is that much rational constraint. But that just shifts the question: what grounds those theories? (There are some remarks in the next chapter, but as we’ll see they are not very unhelpful.)

So where have we got to? Parsons’s picture of arithmetic retains a role for Hilbertian intuition. And unlike an “all-in” holism, he wants to emphasize the epistemic stratification of mathematics (though his remarks on that stratification really do little more than point to the phenomenon). But still, “our view does not differ toto caelo from holism”. And I’m left really pretty unclear what, in the end, the status of the whole web of arithmetical belief is supposed to be.