As I had already discussed, the common binding space (CBS), the substrate of qualia, is three-dimensional. It is for this reason that all living beings, I suspect, are aware of belonging to a three-dimensional space. It seems like this is not a mere coincidence, but has to do with the three dimensional nature of our universe itself.
This begs the question that, if there were a hypothetical universe that has a higher-dimensional space, can there be conscious entities in it with a correspondingly higher dimensional CBS?
If you feel I am jumping the gun here, you are right. We aren't anywhere close to solving the hard problem for our present 3-D universe, so the last thing we need to think of at this point is fictitious universes of higher space dimensionality.
But, it could be quite plausible that there is some kind of logical reasoning (the same kind of logical reasoning that might be used to solve the hard problem in the first place) which rules out CBS's of higher dimensions.
I am less interested in species-specific qualia spaces. For example, we don't know what it is to be a bat. For this reason, I had discussed the concept of platonic qualia spaces, which encompasses every possible qualia space possible by any physical process in our universe according to its known laws. It is possible that, if there are living beings on neutron stars as postulated by Frank Drake, they have access to entirely different subspaces of the platonic qualia space as compared with carbon-based life forms. But the platonic qualia space is a superset of all possible qualia spaces that can occur in a 3-D physical setting.
But I had also postulated that, whatever be the physical process, the CBS itself is 3-dimensional.
But, if we assume the hard problem does eventually get solved, can we come up with a hypothetical universe with physical laws of higher dimensionality which lead to CBS's of higher dimensionality? This would be an "extra-hard problem", so to speak.
Mathematicians would love to be able to visualize four dimensions. They can actually then "see" the beautiful relationship between pi and e given by e^(i. pi) = -1. If only there existed a 4-D CBS!
Note- the dimensionality of the CBS should not be confused with the dimensionality of the platonic qualia space. The latter can be thought of the dimensionality of a vector that gives all the possible combination of orthogonal sensations at any point in the CBS perceived by a conscious entity. It can approach infinity, as in the case of sound. At any rate, the platonic qualia space has a dimensionality much greater than three.