Why Space has Three Dimensions – see Matthew Chalmers, New Sci 28 Sept 2013
That space is 3-D follows from electromagnetic and gravitational theory, both being experimentally validated and both free of quantum theory’s offence to views of reality (Chalmers, NS 28 Sept, p.34).
The reasons are surely much deeper than the inverse square ‘law’ (argument in the Box).
First, accelerated charges emit synchrotron radiation in a beam relative to the inertial reference frame of special relativity, of 3-space plus time.
Second, Einstein’s general relativity gives slightly different answers for light travel times (eg. Earth to Venus) for different accelerated frames – a unique answer comes by specializing to the inertial frame.
Einstein’s strong form of the “equivalence principle” led him astray, in that it took accelerated frames as equally valid, rejecting a unique space frame. His followers like Fock, Weinberg and Logunov have found good reasons for the inertial frame and thus for 3-space being real.
Seeing 3-space as real and higher dimension spaces as mathematical models, thus implies a weak form of the equivalence principle, applying locally, eg. in Einstein’s lift, but not in rotating planetary systems.