Regarding the dimensionality of concepts as implemented in a neuro-computational model using n-dimensional arrays:
We receive visual information in the form of a two dimensional matrix. Each eye's field of perception can be described with two orthogonal polar coordinates; angle and magnitude. With this information alone, all our visual concepts should be flat and two-dimensional. And yet we perceive depth. We take these streams of two dimensional information and add a third dimension; our visual concepts seem to include this dimension. We should therefore expect visual concepts to be represented as three dimensional arrays.
How is this done? Depth perception is deeply intertwined with being binocular; the combination of the information from each eye is what gives us the sense of depth (though mono-ocular people reportedly use other cues, and might provide an interesting contrast). I therefore hypothesize that there must be an area of the brain which observes activity caused by the inputs from each eye and integrates them into a third dimension, and that the transformation from two to three dimensions could be modeled mathematically.
How to test this hypothesis? The bi-hemispheric nature of the cortex suggests an experimental protocol. Since each eye communicates only with its own hemisphere of cortex, the integration of information from each eye must depend on communication between the hemispheres. The differences in simultaneous input received by each eye could perhaps be abstracted into the "depth" dimension. By what mechanism might this occur? Answering this question is the goal of this proposal.
Experimentation might begin by observing the how split-brain subjects perceive depth. If the assumptions above are true, then the severance of the corpus callosum should result in a deficit in depth-perception, perhaps similar to mono-ocular subjects. Experimental testing of this hypothesis should be relatively straightforward. [Added: its likely that this has already can be researched and that this portion could be accomplished by a literature review, for instance see this paper. Also this one, which does not support the hypothesis but perhaps lends some alternative, as well as providing copious references]
Next, comparative studies of split-brain and normal subjects could be undertaken with FMRI in order to identify what brain areas are active in depth-perception tasks. If significant differences are found, a histological study could be undertaken to identify the structures which act as inputs to the depth-perception related areas.
Identifying the way in which these structures interact would ideally result in a model of how higher-dimensional concepts are formed by abstracting from lower-order inputs. This approach could be used in studying abstractions of a fourth dimension as well: changes in input over time.
[Added: The cat paper I linked to above revealed something I didn't know: neurons from the the inside portion of the retinal feild of each eye cross to the other side of the cortex in the optic chiasm. That may shoot down my corpus callosum idea, but it opens another possibility.]