@FBM,
Quote:...it seems to me that it makes perfect sense to say that both are running slower/faster than the other, as long as they remain in their own frames. I don't think it's a good idea to reject a result just because it's counter-intuitive.
You seem to be forgetting one thing, FBM, to wit: Fowler CLEARLY says they are moving relative to each other. Is it POSSIBLE, at all, for both to be at rest, under those circumstances? Notice he does NOT say that they "really are" each moving slower than the other, only that they "appear" to be.
Every physicist I've ever seen weigh in on the matter, starting with Al himself, says that it CAN'T be the case that "each clock is slower than the other."