Few works of philosophy are more perverse than Norman Malcolm's (1959) Dreaming, in which he argues that dreams do not occur when one is asleep and contain no feelings, imagery, sensations, or the like. Malcolm's view, rather, is this: When we wake, we're inclined to confabulate stories of a certain sort. Telling such a story is what we call "relating a dream", but there is no sense in which the dream exists independently of or prior to the story we tell about it, and no sense in which such a story can be evaluated as an accurate or inaccurate description of occurrences during sleep.
Malcolm knew about the then-recent REM research that most people think creates serious problems for his view. He cites and dismisses, on what seem to me flimsy grounds, Dement's research suggesting that reported dream duration on waking matches duration of REM sleep and his very suggestive finding that horizontal movements of the eyes during REM sleep correlate with dream reports of horizontally-salient events (such as a tennis match) and vertical movements of the eyes correlate with reports of vertically-salient events (such as climbing a series of ladders, looking up and down them).
So I don't recommend trusting Malcolm on dreams. But here is an interesting remark of his:
In a dream I can do the impossible in every sense of the word. I can climb Everest without oxygen and I can square the circle (p. 57).So here's my question: What does it mean to say one can "square the circle" in dreams?
Surely Malcolm doesn't mean that the dreamer can actually coherently conceive of a square circle. That, I take it, is straightforwardly impossible. Somewhat differently, can one violate the laws of math and logic in fiction? Can I coherently tell a story in which 2 + 2 equals 5 (like I can coherently tell a story in which pigs fly)? I'm not particularly well read in the metaphysics of fiction, but my impression is that few philosophers of fiction would grant that -- though if we do grant it, we might be able to use it (with some additional assumptions) to give sense to the idea of squaring a circle in a dream.
More likely, Malcolm is expressing the idea that we can reach incredibly stupid judgments in our dreams, even baldly contradictory judgments. I have also long thought this (see section v. of this essay).
If we accept this view, it gives more juice to the dream skeptic than she is usually accorded. For if this right now might be a dream, then mightn't also my thinking be so baldly contradictory that I can't even trust my simplest-seeming judgments? The dream possibility calls, then, not only sense experience but also reasoning into doubt. If we allow ourselves the assumption that a faculty frequently unreliable is not to be trusted, there would be no hope that I can rely on my reason to establish the hypothesis that I actually am awake, or indeed anything else.