What shape has mindspace?

If a mind is a topological space equipped with a subset, what sort of mind would the set being full imply?

That I'm not sure, as I haven't worked this out in much detail. I just sort of have a vague mathematical intuition that it might be the right sort of way to model it (n.b. I dropped out of a math phd after 6 years to do a startup, if that's some rough guide to how much to trust my mathematical intuition).

For what it's worth, here's some notes I found that I wrote about this a while ago. I make no promises that any of this makes any sense or that I would still agree with any of it, but it is what I wrote about it a while ago:

Mathematical Foundation of Phenomenology

So that phenomenological complexity classes are applicable to as many universes as possible, including our own, it has a rigorous mathematical foundation that makes as few assumptions as possible and easily translates into the standard language of phenomenology. That said, it is not a theory of everything, and so supposes that

• the universe is made of stuff in configurations called states that are related to each other by causation,
• mathematics can be applied to stuff, states, and causation,
• and states can be partially ordered by causation.

Let process denote a set of states partially ordered by causation. Processes include simple physical processes modeled by atoms and quarks, stochastic processes like Brownian motion and waves, mechanistic processes like levers and clocks, phenomenological processes like cats and humans, social processes like organizations and friendships, and a universal process that includes all states and which we refer to as the universe, reality, or the world. Every process is a subset of the universe, including the empty process that contains no state.

We can then construct a topological space, called process space, on the subset processes of the universe where

• every process, including the universal process and the empty process, is a member of the topology,
• the union of processes is itself a process,
• and the finite intersection of processes is itself a process.

Process space, being a topology, automatically gives us a lot of constructs to work with. Each process has an interior defined as the process containing itself and all the processes within it. We can further define filters in process space as the partial ordering of processes by subset, and then the closure of a process will be all the processes within the limit of the filters with the process’s interior as their base. These definitions are respectively analogous to subject, experience, and context in the standard terminology of phenomenology, but are strictly ontological and lack confounding teleological meanings. We’ll use these ontological terms in the remainder of the introduction for precision but switch the more common terminology in the dialectic for readability.

Processes may contain within them partial information about their filters in their partially order set of states. This is possible because partial ordering on filters

Phenomenology is the study of process filters, and the filters on a process are closed under finite intersection, which is to say that the commonalities of any two experiences is itself an experience, and upwards closed under subset, which means any part of an experience is itself an experience. Then we can understand the largest filter on a process’s interior, which is to say the filter that cannot be made any larger and still have the process’s interior as its base, as the principal ultrafilter on the process’s interior or simply the process’s ultrafilter. It trivially follows that a process’s ultrafilter contains every filter on the process as a subset, thus a process’s ultrafilter can be understood as the totality of a subject’s experience.

Thus far our theory is , and so when we talk about a process’s ultrafilter we are talking about all the experiences a subject might or will have, depending on the nature of causality. Yet each filter that is not itself the ultrafilter has supersets, which we might call “future” experiences. Since we are ourselves processes embedded in the topological space, at any “moment” of experience we have “future” experiences, and so when we talk about a process’s ultrafilter it’s useful to have a notion of the ultrafilter in time for universes like our own where the state of stuff is grouped in partially ordered subsets of the power set of state sets that admit a notion of “before” via subset, “after” via superset, and “now” by choosing an arbitrary set as reference point.