The Algebra of Views & The Big Picture
It seems that all of this is rolled out as follows:
0) 1) 2) 3) There is first a unified outlook (such as God) which unfolds the perspectives that are the basis for Structure, for the DivisionOfEverything. We may think of this unfolding as an operation [AddOne +1] which keeps reinterpreting the whole as an additional perspective. When there are three perspectives, then the structure for the original outlook is complete: it can understand, come to understanding, and be understood. However, the operation +1 continues.
4) 5) 6) Therefore a new outlook awakens and finds itself as such within the structural situation unfolded by the original outlook. We may think of this as a "godlet" which may not be God, but is otherwise in the situation of God. There is now a disconnect between Structure and Activity. Structure may or may not channel activity. Activity may or may not evoke structure. The feedback between structure and activity may be thought of as an operation [AddTwo +2]: the evoking of structure is linked to the arisal of activity. We may think of the godlet as a perturbation that opens up angles: Representations upon the whole, and Topologies from out of the parts. I think that this is where the "algebra of views" is defined. The give and take between activity and structure introduces a slack which allows one to take up a perspective, thus integrating whole and parts.
7) Then the new outlook comes to understand itself with regard to the original outlook as a perturbation of an ideal outlook that links both outlooks. All three outlooks are characterized by their three-cycles: taking a stand, following through, reflecting. And these rotations may be thought of as an operation [AddThree +3]. I think here is where the dynamic languages of life come into play: argumentation, verbalization, narration. I suppose they are expressions of the "algebra of views". Here the ideal outlook serves as a mediator which allows us to localize the slack so that we know where it is within a three-cycle. This makes the algebra definite.
8=0) Then the new outlook understands itself as subordinate to the original outlook. At the core of the new outlook is always the original outlook which went beyond itself and thereby generated the new outlook. Everything is always collapsing back into the original outlook. The views of the new outlook and the original outlook coincide by way of that collapsing.
This is extremely helpful for me because it places the "algebra of views" within the big picture. It suggests that the algebra of views becomes defined with the divisions of everything into four, five and six perspectives. And that its applications through argumentation, verbalization, narration arise with the division of everything into seven perspectives. And, finally, the coinciding of views is related to the collapse of structure, which is perhaps the key point about mathematical systems in general. It's the collapse of structure which makes mathematics interesting.