Well then, speaking as an Extreme Platonist can you tell if the real line well-ordered? Or equivalently, are there infinite sets so big that one cannot tell if one is larger than another? Or equvalently, does there exist an unmeasurable set of real numbers (outer measure 1 and inner measure 0 for example.)
These are serious questions for the Platonist position.
With the conventional definition of "well ordered" (every non-empty subset has a least member) then the reals are not well ordered. I'm sure you know the proof as well as I do.
But I seem to have stupidly failed to grasp your point. Why is this an important question? The important question, to me, is rather the question whether the conventional definition of "well ordered" makes sense. Which has very little to do with Platonism.