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I'm interested to try to use category theory to model the composition of perspectives. I wonder to what extent that's already been done. Somewhat related is Gilles Fauconnier's important work on Mental Spaces.
For example, consider a lost child in an airport. An oblivious child doesn't even realize that they are lost. A foolish child goes looking for their parents. A wise child realizes, "I am the child and they are the parent. They are supposed to be looking for me!" And so the wise child goes to wherever their parents will most easily find them.
The intelligence of the children is given by the length of composition:
The oblivious child entertains: The child's view of the parent's view of the child's view. ("My parents hear me.")
The foolish child entertains: The child's view of the parent's view of the child's view of the parent's view. ("My parents need me to understand that they have lost me.")
The wise child entertains: The child's view of the parent's view of the child's view of the parent's view of the child's view. ("My parents should be concerned that I am wondering if my parents know what I am going through.")
The wise child's chain of perspectives is extraordinary in that it allows the child and the parents to coordinate their actions (the child will go where the parents will most easily find them) without any communication but simply the knowledge of asymmetry (who is the parent, who is the child, and who should be looking for whom). For me, personally, I believe it's a chain of the maximal length that I am capable of.
I'm not sure how to define "perspective" although I have thought a lot about perspectives. For me, a synonym would be "point of view". I suppose that it relates to the knowledge that is available in a given circumstance. As such, it may also depend on the observer taking up the perspective.
It's also important for me to be able to model "stepping-into" a perspective and "stepping-out-of" any perspective. Stepping-into a perspective means that I am engrossed in a perspective and no longer cognizant that I happen to be in a particular perspective. Stepping-out means that I am not entertaining any perspective but cognizant of the different perspectives that I may take up. We are subjective when we are stepped-in and objective when we are stepped-out. In a healthy discourse, participants alternate unpredictably as to who is stepped-in and who is stepped-out, and there is even a sense of a fluttering spirit over the discourse. I'm wondering if there is an adjoint type of relationship as in "there exists a perspective in which we know..." and "for all perspectives we know..."
It's not clear if composition of perspectives is associative. I'm not sure if category theory is ultimately the right mathematical framework but I'm curious to try. Success would be great and failure might help discover a more appropriate framework.