It has been clear for a very long time-- at least since I was a student in the 1970s-- that simple possible-worlds treatments do badly at accounting for EPISTEMIC modality of the sort in question. (This despite the fact that, not long after the beginning of modern possible-worlds semantics in the 1960s, Hintikka described a possible-worlds semantics for epistemic and doxastic notions in his influential book "Knowledge and Belief." I think he was explicit in saying that the theory was an idealization, not applicable to mathematical ignorance without refinement.) (((Vocabulary: "Epistemic" logic is often construed broadly, as covering knowledge and a variety of related notions. When knowledge is being distinguished from (mere) belief, "doxastic" refers to the belief operators.))) It is, I think, much harder to model MATHEMATICAL ignorance in a possible-worlds framework than it is to give an account of the sense in which, e.g., water MIGHT (have turned out) not (to) be H20. Kripke ("Naming and Necessity") pointed out that the latter is an epistemic "might" (and not the "metaphysical" possibility he was primarily concerned with), but it seems possible to define THIS kind of epistemic possibility: roughly It is metaphysically possible for their to be someone whose evidence for a statement of the same logical form as "water is H20" is similar to our evidence (before modern chemistry) for "water is H20" even though their statement is false. This won't work for mathematical statements if the "evidence" is allowed to include knowledge of the truth of axioms from which the statement follows logically. Many approaches to this problem have been suggested; none to my knowledge have been worked out in adequate detail: (1) Early workers in RELEVANT (or relevance) logics often gave, as a motivating hope, the possibility (so to speak) of developing an epistemic logic based on them which would tolerate ignorance of mathematical and logical truths. (Since the Routley-Meyer semantics for relevant logics makes use of "worlds" -- they preferred to call them "set-ups" in their early papers -- in which logical truths fail, this suggestion could be seen as connecting to the idea of partial worlds, though their set-ups could be inconsistent as well as incomplete.) (2) Other suggestions about the use of partial or inconsistent structures instead of full, classical, models as possible worlds have been made. One of Hintikka's compatriots (sorry, I've forgotten the exact reference and will try to give it in a supplementary post if no one else beats me to it) published a paper in an early (1970s) volume of the "Journal of Philosophical Logic" proposing that "surface" models be used. This is a notion derived from Hintikka's work on Distributive Normal Forms: it would allow "possible worlds" in which sentences inconsistent by First Order Logic are true, provided that the proof of their inconsistency exceeds some measure of quantificational complexity. (Sorry to give such a vague account of it, but this letter is long enough as it is! If you are not familiar with Hintikka's work and are wondering whether you would be interested enough for it to be worth following up, think "Herbrand expansions, complexity of.") (3). I think the problem is an open one, and perhaps-- given the use of epistemic logics in AI-- an important one. Note that approaches (2) and (3) above, and probably even (1) (though Meyer and Routley themselves were not Platonists) are consistent with quite extreme forms of Platonism. I would expect a satisfactory solution to be so consistent. --- Allen Hazen Philosophy Department University of Melbourne