I’m still working away on Parsons’s book — and I’m rather stumped by his claims about “intuitive knowledge”. One worry is this: he introduces the notion in cases where we acquire propositional knowledge by, as it were, “just seeing” e.g. that “|||” is the successor of “||”. But he fairly rapidly wants to extend the notion of intuitive knowledge so that it is preserved under some, but not all, logical inference, and some but not all applications of arithmetical induction. And I just can’t see what the constraint on the notion is that rules some cases in and others out — for that constraint certainly isn’t implicit in the cases which introduce the notion in the first place.
Well, be all that as it may. For light relief — and to see if any sideways light can be thrown on Parsons’s on intuition more generally — I’m reading Marcus Giaquinto’s Visual Thinking in Mathematics (OUP, 2007). The first few chapters already show that, unsurprisingly, the book is written with Marcus’s customary clarity and good sense. I’ll report back in due course as I read more: but already I’m having to backtrack a bit and slightly rethink things that I earlier wrote on Parsons.