I've been trying to make good sense of the rest of Parsons's chapter on intuition, and have to confess failure. We might reasonably have hoped that we'd get here a really clear definitive version of the position on intuition that he has been developing for the better part of 30 years; but I'm afraid not. Looking for some help, I've just been rereading James Page's 1993 Mind discussion 'Parsons on Mathematical Intuition', which Parsons touches on, and David Galloway's 1999 Philosophical and Phenomenological Research paper 'Seeing Sequences', which he doesn't mention. Those papers show that it is possible to write crisply and clearly (though critically) about these matters: but Parsons doesn't pull it off. Or at least, his chapter didn't work for me. Although this is supposed to be a pivotal chapter of the book, I'm left rather bereft of useful things to say.
Sec. 27, 'Toward a viable concept of intuition: perception and the abstract' is intended to soften us up for the idea that we can have intuitions of abstracta (remember: intuitive knowledge that, whatever exactly that is, is supposed to be somehow founded in intuitions of, where these are somehow quasi-perceptual). There's an initial, puzzling, and inconclusive discussion of supposed intuitions of colours qua abstract objects: but Parsons himself sets this case aside as raising too many complications, so I will too. Which leaves the supposed case of perceptions/intuitions of abstract types (letters, say): the claim is that "the talk of perception of types is something normal and everyday". But even here I balk. True, we might well say that I see a particular squiggle as, for example, a Greek phi. We might equivalently say, in such a case, that I see the letter phi written there (but still meaning that we see something as an instance of the letter phi). But I just don't find it at all normal or everyday to say that I see the letter phi (meaning the type itself). So I'm not softened up!
Sec. 28, 'Hilbertian intuition' rehashes Parsons's familiar arguments about seeing strings of strokes. I won't rehash the arguments of his critics. But I'm repeatedly puzzled. Take, just for one example, this claim:
What is distinctive of intuitions of types [now, types of stroke-strings] is that the perceptions and imaginings that found them play a paradigmatic role. It is through this that intuition of a type can give rise to propositional knowledge about the type, an instance of intuition that. I will in these cases use the term 'intuitive knowledge'. A simple case is singular propositions about types, such as that ||| is the successor of ||. We see this to be true on the basis of a single intuition, but of course in its implications for tokens it is a general proposition.A single intuition? Really? If I'm following at all, I'd have thought that we see that proposition to be true on the basis of an intuition of ||| and a separate intuition of || and something else, some kind of intuitive (??) recognition of the relation between them. What is the 'single' intuition here?
Or for another example, consider Parsons's wrestling with vagueness. You might initially have worried that intuitions which are "founded" in perceptions and imaginings will inherit the vagueness of those perceptions or imaginings (and how would that square with the claim that "mathematical intuition is of sharply delineated objects"?). But Parsons moves to block the worry, using the example of seeing letters again. The thought seems to be that we have some discrete conceptual pigeon-holes, and in seeing squiggles as a phi or a psi (say), we are pigeon-holing them. The fact that some squiggles might be borderline candidates for putting in this or that pigeon-hole doesn't (so to speak) make the pigeon-holes less sharply delineated. Well, fair enough. I'm rather happy with a version of that sort of story. For I'm tempted by accounts of analog non-conceptual contents which are conceptually processed, "digitalizing" the information. But such accounts stress the differences between perceptions of squiggles and the conceptual apparatus which is brought to bear in coming to see the squiggles as e.g. instances of the letter phi. Certainly, on such a view, trying to understand our conceptual grip here in terms of a prior primitive notion of "perception of" the type phi is hopeless: but granted that, it is remains entirely unclear to me what a constructed notion of "perception of" types can do for us.
Sec. 29, 'Intuitive knowledge: a step toward infinity' Can we in any sense see or intuit that any stroke string can be extended? Parsons has discussed this before, and his discussions have been the subject of criticism. If anything -- though I haven't gone back to check my impression against a re-reading of his earlier papers -- I think his claims may now be more cautious. Anyway, he now says (1) "If we imagine any [particular] string of strokes, it is immediately apparent that a new stroke can be added." (2) "Although intuition yields one essential element of the idea that there are, at least potentially, infinitely many strings ... more is involved in the idea, in particular that the operation of adding an additional stroke can be indefinitely iterated. The sense, if any, in which iteration tells us that is not obvious." But (3) "Although it will follow from considerations advanced in Chapter 7 that it is intuitively known that every string can be extended by one of a different type, ideas connected with induction are needed to see it." We could, I think, argue about (1). Also note the slide from "imagine" to "intuition" between (1) and (2): you might wonder about that too (Parsons is remarkably quiet about imagination). But obviously, the big issue is going to come later in trying to argue that ideas "connected with induction" can still be involved in what is "intuitively known". We'll see ...
Finally, I took little away from Sec. 30, 'The objections revisited', so I won't comment now.