Intuition of Objects vs. Holism

In a previous post I wondered what the role of intuition of quasi-concrete objects like stroke-inscriptions really was in Parsons' overall epistemology of mathematics. After finally finishing the book, it seems that one clear role, at least, is Parsons' objections to holism of the sort familiar from Quine and championed in more detail for mathematics by Resnik. In the last chapter of the book Parsons makes this point:

Intuition does play a role in making arithmetic evident to the degree that it is, in that there is a ground level of arithmetic, not extending very far, that is intuitively evident. Furthermore, the objects that play the role of numbers in this low-level arithmetic can continue to do so in a more full-blooded arithmetic theory.

After noting that logical notions allow this further extension, he insists that

the role of intuition does not disappear, because it is central to our conception of a domain of objects satisfying the principles of arithmetic ... an intuitive domain witnesses the possibility of the structure of the numbers (336).

Here, then, we have a definite epistemic role for intuition of objects. It helps us to explain what is different about arithmetic, or at least the fragment of arithmetic that is closely related to these intuitions. (In chapter 7, this fragment is said to not even include exponentiation, so it fars fall short of PRA.)

While this objection to holism is quite persuasive, Parsons is at pains to emphasize how modest it really is. He offers some additional discussion of the implications for set theory, but the book seems primarily focused on what distinguishes arithmetic from other mathematical theories. It is an impressive achievement that I am sure will frame much of philosophy of mathematics for a long time.