Does Flatland exist according to modal realism, since it is a possible world?
Yes, according to Lewis. In Lewis' modal realism, to say that some statement R(x) is possible dictates that a possible world exists in which R(x) is not only possible but actually true. This stance is called alethic modality (intimately related to epistemic modality in philosophy of language). Note that Lewis does not claim that for each statement R, a possible world exists in which R holds (consider a world in which R and not R are true at once - such a world cannot exist in an alethic modal sense if we are interested in retaining the Law of Noncontradiction.
No, according to Lewis!
According to Wikipedia:
Lewis believes that the concept of alethic modality can be reduced to talk of real possible worlds. For example, to say "x is possible" is to say that there exists a possible world where x is true. To say "x is necessary" is to say that in all possible worlds x is true. The appeal to possible worlds provides a sort of economy with the least number of undefined primitives/axioms in our ontology.
I take this to mean that in some possible world, everything's exactly the same as this one except that I'm right-handed. (I happen to be left-handed in this world.)
This is perfectly sensible. My handedness could easily have been different. [Although I would actually dispute that. Being left-handed branded me an outsider as a child, needing special scissors in grade school, having problems with handwriting, etc., thereby affecting my adult personality. So I would argue in general that the notion that we can "flip the truth value" of a large collection of propositions is false. Everything is so interrelated that we can not keep everything else the same and simply change the truth value of even a contingent truth. I'd regard this as an argument against modal realism. But that's outside the scope of this discussion. Here, we're accepting modal realism and the idea of flipping the truth values of propositions in order to form possible worlds.]
What about the proposition that 2 + 2 = 4? I think most people would say that this is a necessary truth: it must be true in all possible worlds. There does not exist a possible world in which 2 + 2 ≠ 4.
Now, consider the proposition P = "The dimension of the universe is 2; and the universe contains living beings with digestive systems."
Is P necessary, like "2 + 2 = 4?" Or contingent, like "The person who runs the handle user4894 is left-handed?"
I claim the dimension of a universe that supports life can not possibly be two; for the topological reason that in two dimensions, a digestive system necessarily disconnects a being. It is a necessary truth that the dimension of a universe supporting living beings who have digestive systems must be greater than 2.
Therefore Flatland can not be a possible world in the sense of modal realism; since two-space does not support beings with digestive systems. Proposition P is false in all possible worlds.
There is no two-dimensional universe containing beings who eat and excrete.