The issues noted in my “Eight Views of Mathematics” have of course been noted by other people before (after all, I was talking about the programs of Field, Steel/Woodin/Martin et al, and Dummett). However, it seems that at least the second one of these questions has been somewhat overlooked in past. Some people (I don’t recall who off the top of my head) refer to the first position as “anti-realism in ontology” and the third as “anti-realism in truth value”, but they don’t seem to have a name for the middle one. Similarly, Mark Colyvan (on p. 2 of his The Indispensability of Mathematics) talks about “metaphysical realism” and “semantic realism”. I suppose I’m tempted to call the third a sort of question of “epistemological realism”, or “realism in evidence” or something of the sort.
The first question is about what objects mentioned in mathematical statements exist, the second is about what mathematical statements we can know, and the third is about what mathematical statements are true or false. Dummett seems to want to link the first and third question. But it seems that most people try to assimilate the middle question to one of the other two. I suppose that’s where it falls to me to defend my position that we can know (or believe or accept) mathematical statements much beyond ZFC without being committed to mathematical objects, or to the claim that all mathematical statements are either true or false. (Though I’m probably agnostic about that last claim, rather than opposed.)