Many thanks for your personel review, Prof. Bob!
Indeed, one could easily draw a line of thoughts from the abolished Limbo, via the abandoned ether and Plato's forgotten „order of heavens" to the hierarchy problem.
A topologist, could take a bundle (F) of hyperbolic spaces with some handles and loopholes in it, apply the Fourier transform, arrive in 6D and - wait for the babies.
A geameter exists already there,- in Plato's ideal world, filled with (Maxwell's and Faraday's) interfering lines of forces, forming triangles over and over, entangled towards manifolds of all possible higher order.
Didn't Klein tell, that all geometries and topology are merging in higher dimensions?
Good question; - can we realy explain „dimensions" with "degrees of freedom" only ?
I guess they are necessary to avoid Hilberts trap and mix up a beer mug with ...