Argument For the Primacy of 3D Relations

Below is a list of number of elements and possible relations between those elements using the above formula:

Now, given that we have an awareness of perception (i.e. cognition can cognize itself), we may draw our attention to the above relations and form secondary relations or associations amongst those relations. When we do the counting for the above sets, an interesting thing happens. Below is the same list repeated with the numbers of secondary 'relations between relations' added in:

1 - 0 - 0

2 - 1 - 1

3 - 3 - 3

4 - 6 - 15

5 - 10 - 45

6 - 15 - 105

7 - 21 - 210

etc.

Now add tertiary relations between relations between relations:

1 - 0 - 0 - 0

2 - 1 - 1 - 1

3 - 3 - 3 - 3

4 - 6 - 15 - 105

5 - 10 - 45 - 990

etc.

Ok, you get the picture. For all elements below n=4 we get a simple repetitive count. At 4 and above elements, however, we get a combinatorial explosion of relations which carries out to infinity! Remember that 4 is the number of elements required to enclose 3 dimensional space. It appears to me that:

3D dimensionality is the minimum manifold which allows recursively infinite and non-degenerate extensibility of relationship.

As the minimum manifold of relational extension, 3D thus forms the simplist underpan upon which to hang a cognitive universe. Now for a bit of interpretation.

For the above degenerate cases of 1, 2, and 3 elements, no new relational information accrues by doing higher order expansions. At 4 and above, however, each round of relational comparison adds extra distinctions. It must be pointed out that each level of expansion represents a higher order of abstraction. One might be tempted to also do relations between elements on different levels. One can certainly do so. In fact, I think, this is one of the great sources of error which is inherent in the semiotic process. By doing so, one is actually comparing oranges to apples and confusing types. It is my belief that this is precisely what art and intuition do. They perform a 'semiotic short circuit' between logical types. In itself, this can can be useful and perhaps gives cognition a quick exit from the box, so to speak.

Another thing to notice is the information content at each level. It is paramount to keep in mind that information always has a spatio-temporal cost. It may appear that we are getting information for free in our expansions above, but such is not the case. Each combinatorial relational expansion only conveys the full informational complement of the original set of elements. The information is simply rearranged and distributed across those relational bridges at each level. Each successive level redistributes the same information across progressively finer and finer distinctions. Each level also has a higher representational load i.e. the informational structures required to fully represent the relations on that level require more bits if you will. The temporal costs should be plainly evident also.

Thus we see that while cognitive relational expansions are in principle infinitely extensible, there are actually real-world limits to the process. It may be that connective thresholds in our cognitive neural apparatus form these limits. The higher order expansions thus would tend to smear out into generalities of perception. This may in fact be the real beauty and power of cognition - the ability to grow, flower, bear fruit, and return to from whence it came.

I believe there is real 'semiotic gold' in the above observations. The application to neural net theory is obvious.

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