I just love a comfy, warm bed at the end of the day. However I may run and shout during the day, I always seem to find myself back in the cave of Hypnos, straddling the banks of the Lethe.
The mind has a "shape," although its would be "full form" is never glimpsed nor felt. But it is known topologically all the same, by the paths in and between its regions. One may find circle-like paths, loops which may amplify into themselves or route freely into other paths, and the difference can be thought of topologically. This is like Mandelbrot's "ramification" of a point, the number of paths leading from a point.
It should be no surprise that the topology of the mind resembles that of fractals. I am often aware, while heading down a more or less straight path, of a multitude of possible roads-not-taken. It is true that at any time, they do not have the infinite depth and recursion, and if we could picture the whole shape, it would not exhibit invariance across an infinity of scales. Nonetheless, any path noticed can be taken by effort or whim, or whatever it is that allows us to guide cognition. Some lead back into others, but there is no obvious limit on the branches from a particular thought-place.
It is as if the structure were already present, and we simply wandered it, charting its curves. Of course, we are constructing and permuting structures as we go. The conditions are indistinguishable "from the inside." With this picture in mind, it is not hard to imagine the cognitive topology as continually reaching towards a fractal shape, though in the long run it can only permute itself into itself, seeming to reach ever towards different "limit shapes"; its capacity for growth of mental structure is limited from below by the smallest molecular structure employed and above by the constraints of the body's total size.
Who knows? It could very well be that even one body can host enough growth that it need never "overwrite" or modify its substrate in a typical human lifetime. I'm not equating cognitive paths with bodily structures, but I do assume that a given bodily structure allows for only so many possible cognitive configurations; there seem to be natural constraints.
Perhaps the most familiar paths feel more solid, perhaps they are comfortable. I feel myself often at familiar junctions, with knowledge of where certain obvious routes lead. Some places feel more free than others.
I have come to believe what is perhaps an ecological equivalent of Maturana's claim about life and cognition. It is that the topology of cognition necessarily mirrors the topology of the environment. Of course, this is necessarily the case only when the organism persists for extremely long periods of time ("...the probability goes to 0 as t goes to infinity").
I am not positive this means that the overall cognitive structure must resemble the topology of the planetary landscape to a creature, but we can be certain that a certain implication must hold: If the environment is a compact space, so is the cognitive system.
The mathematical proof is trivial: The body is always a compact space; the mind has nothing but the body and its environment to grow upon.
In nature's organizational topology, "compactness" corresponds to a system which is clearly finite, given some perspective: The sum of all parts of the space (= the space) can be measured by a finite number of discrete acts of measurement. In the day of satellites and jets, we conceive of our planet as a finite space. We can calculate this by imagining a sphere around our body, an intuitive representative of our awareness, connection, and nearness to objects. It is a fuzzy object, never exactly certain, but discovered to some degree, often with various dependencies, by each organism via organismic equivalents of Monte Carlo methods.
Factoring out time, it's phsyically conceivable to touch one's sphere to every part of the planet's surface, whether including liquid surfaces or not (in practice, every relevant or passable part of the surface). Either way, the ocean would be a challenge, but it's hypothetically possible. We can imagine a finite amount of configurations would in this way "cover" the accessible planetary sphere. One could visit every human-defined "place" in the environment in a finite number of steps.
In reality, many places would be impossible to enter safely for legal reasons. Legally, those are not technically open places to you; you cannot go there; visiting them cannot be inside your experience. Considerations like this alter the constructed topology, and one intuits "holes" in the envioronment. Considerations involving gravity and velocity can "cut off" the edge of a balcony, for instance (unless you have a parachute or a squirrel suit perhaps - but these flying devices also have their apparent environmental limitations in their functions to allow persistence). One often has the physical freedom to act against such restrictions, of course, but these people tend not to persist in thes relatively free configurations.
In this way, modern industrial states have become as a force of nature. The algorithm has been instilled since childhood: Trespass the law enough, and the consequences will be inevitable and forceful: Detainment and physical punishment. Creatures who choose certain paths tend not to persist in their former society. (Although they may enter new societies persistently in places of detainment).
It's worth mentioning that for an extremely small organism, even small regions of the planet are virtually infinite, not compact when speaking topologically. This is true for small creatures, and for many people in a social sense; we will never practically run out of people to meet.
It's also true of the globe, on average, in a practical sense. The living structure of the surface is extremely important for humans - plants can be climbed or block passage, seeming to add or delete paths. If one were to attempt to ascertain the condition of the globe, taking the shortest possible path, one would inevitably find that things had changed by the time one returned to the starting point. Plants have grown or died, animals have moved around, made new homes, had children. The only way this could be avoided is to start from a completely dead environment - by definition impossible for a human.
Abstractly, it is the old problem of having a carpet too big for the room: You can never get every corner down without lumps somewhere.
This is why beliefs, not only about "what is out there," but about the consequent constraints on safe, healthy, or possible action are so influential on a person's life. They can turnt the planet into a small town or humanity into a few hundred. They can turn the world into a football game, with star players and sore losers. If one believes oneself stuck in a certain societal position, one either tries to leave or immediately alter the situation, or agrees to live within the believed constraints, to acknowledge their reality. We compactify the spaces of our daily lives to make decisions tangible and urgent.
It perhaps does not immediately follow from the compactness of the body, but it's just a leap away to intuit that no cognitive path can be built but that which starts from some base point, and none can ultimately farther away from it than is allowed by physical constraints. One can only build off experience, only take a path so far before it turns back on itself. This zenith, "the farthest you can go before you start coming back" implies a symmetric reverse path; if this trajectory repeats, the experience may feed back, gradually altering, but unwavering in its essentially circular topology. No matter how quickly you run, you can' t achieve escape velocity.
One thing that makes human life seem infinitely varied is the apparent non-compatcness of the social space, the fact that it is not biologically conceivable that one could account for every living human on Earth. Another is that the mind can take such a variety of shapes, find so many ways to operate in its world. Its only necessity for persistence is that its mind slides often enough over some understanding of "actual" aspects of its environment. The rest is negotiable.
Perhaps a compact space is not enough for some, claustrophobic or distasteful. Chaotic, wet, dirty, gross, savage. What fuels the zealous dreams for space travel among our people? Cold war propagandists and Hollywood producers wished they had the ability to craft a dream so beloved; they merely tapped into an old cultural structure.
I like Earth. No place in the observed universe like it. There are insane problems, it is true. But surely, any traveller in the sea of space would eventually get homesick and circle back? What beautiful nebula, what awesome star, what grand and distant, if lifeless, galaxy could squish the nagging anxieties back? Leaving earth permanently (and surviving) would not only be a miracle of ecological engineering (alternately, an irreversible theft from Earth's living sphere), it's fighting millions of years of emotional evolution. It's self torture.
What solution is there? Detachment? Exploration of undreamt psychological realms? Transcendent self-disintegration? Deep space as isolation tank? Works great for sci-fi. Works the best for tragic sci-fi. Besides, insanity and nirvana work just fine on Earth.
Let's try another route: Roam the abyss with a partner or commune of like-minded individuals?
So one is alone in a confined island (compact, however large) of survivable space with other humans, with their own dreams, desires, plans, theories, morals? Is science and exploration powerful enough to draw people altogether away from the town square? What of political economy (who will do what work?), conflicts of interest, clashing beliefs about course or research focus? What of governance?
Perhaps a single like-minded syndicate seems to much like a societal monoculture. What then, does one build a relativistic city? One only amplifies and intertwines the political complications, runs high risk of a larger (and repressive) monoculture.
Okay then, how can we fix this? Send many cities or nations in separate vessels, with their independently developed cultures and attitudes?
Golly, can't see how that could go wrong. Immediately, questions, disagreements, heated arguments: should we allow them to freely board and leave our vessel? We have surplus and they a need, do we use their desperation to gouge them?
At some point one's forced to beg the question - just how much of Earth are you willing to take with you in order to get away from Earth? The only demons out there are the ones you take with you.
Sometimes I wonder if it's a good thing that there is so much non-human, uncontrolled noise in our planetary lives to dampen our pained vibrations.
We can't reach for the stars without reaching inside ourselves. I almost wish such a trite sounding phrase didn't summarize the situation so efficiently.
What better way is there to progress, if that is possible, than to start, right now, in each possible moment, to heal the wounded human tissue that huddles in clumps on the humid, wet rock, Earth? To embrace the world where we can, with all its sticky consequences? To attempt as soon as possible to re-arrange our society, state and economy into something which is not only ecologically sound, but, dare I say, pleasant?
Human worlds remain necessarily compact topological spaces, not only geographically and societally, but psychologically, emotionally.
But this does not necessarily entail that it is finite, limited, scarce. The geometry of life and society is necessarily fractal to continue itself; it has always been this way, that is exactly how it got here. Wishing to keep it between rectangular lines and bound by hierarchical control circuits is always a paradox to start. [The bourgeois dream is wet]. This organizational form never quite gets off the ground, despite its enthusiasm, its fervor. Always, the humans are in there, living components, sliding fluidly over the rigid organizational structure like they do the cement-asphalt-steel skeleton which is the symbol and substantiation of its persistence.
Like those goddamn tenacious rose bushes. My height in half a year, sheesh. Spilling petals all over the place and everything.