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Is 3+1 spacetime as privileged as is claimed?
Regardless of your favorite theory of how many dimensions the universe has in total, the universe seems to have a deep preference for displaying three fully interchangeable large-scale spatial dimensions (plus time) within any given frame of reference.
But why? That is, has anyone ever come up with a persuasive argument for why the number three is or is not the required number of large-scale spatial dimensions in a universe?
This question from way back in November 2010 appears related, but after reading through it I'm pretty sure it is only pointing out that the known laws of electrostatics have three spatial dimensions built into them, which is most certainly true. But universes with fewer or more than three large-scale spatial dimensions would presumably have different force rules reflecting their different geometries, so this kind of analysis only shows self-consistency within a universe.
(I should also warn responders in advance that while I respect the right of some theorists to suggest that it is three "because our universe evolved that way out of a fractal multiverse," I also respect my own position that such statements are equivalent to saying "We haven't the foggiest idea why." An answer along those lines would at the very least need some powerful anthropic principle support, e.g. a proof or near-proof that universes with less or more than three large-scale spatial dimensions would have extreme difficulty supporting life.)
Answer
A few quick references before you close the question:
There's a rather technical discussion on what's special about four dimensions on the Math Overflow (way over my head!).
The article I was thinking of is actually on Wikipedia. This picture from the article:
succinctly explains why our space is 3+1 dimensional.
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