This is Part 4 of the Babble and Prune sequence.

In the previous parts, I described the brain's thought-generation process as an adversarial learning system between Babble - which generates low-quality content - and Prune - which filters for high-quality content.

My primary motivation for understanding this system is to solve writer's block. On some days, I achieve a state of flow and write two thousand lucid words in two hours, which somehow coalesce a previously disconnected body of thoughts. On others, I give up after one or two navel-gazing abortions. These irregularities are a sign that my model of Babble and Prune is insufficiently predictive.

Babble and Prune are part of the picture, and a conscious effort of systematically relaxing the lower Gates of Prune has helped me produce more material. However, there's at least one important point which is entirely missing in this model: the choice of subject matter, the focus of one's attention towards which Babble and Prune approach.

Today I will fit Babble and Prune into another piece of the puzzle: Circumambulation - walking around the truth, spiraling towards it.

Truth is a process of successive approximation. More precisely: truth is a process of circumambulation. Deep truths are not mere point masses on a line to be approached with binary search or gradient descent. They are sprawling manifolds in high-dimensional space, and to map out such a Titan requires patience, false starts, and approach from many directions.

Circumambulation is the underlying directive of Babble, the process of circling around the holes in your understanding to pin down their shape and fill them in with substance.

1. Circumambulation

From Jung:

I began to understand that the goal of psychic development is the self.
There is no linear evolution; there is only a circumambulation of the self. Uniform development exists, at most, at the beginning; later, everything points toward the centre.
This insight gave me stability, and gradually my inner peace returned.

This quote is accompanied by the lovely picture of one of the labyrinths found in many cathedrals. Classically, cathedrals are built in the shape of a cross, e.g. Notre Dame:

At the center of the cross (X marks the spot) is the labyrinth. Walking the labyrinth is a ritual reflecting on the nonlinearity with which we circle the truth.

Our imagery for reality is deeply tied to our spatial reasoning. Metaphors like map and territory, thingspace, and pendulums make instinctive sense by tapping into this connection. New knowledge is produced by honing in on holes in the map, and then filling them in by surveying the territory.

The map can be a very twisted and high-dimensional thing full of holes and jumps, and Babble is a biased random walk like simulated annealing on this space.

Babble is the process of randomly walking around the map, honing in on beacons, holes, and discontinuities. These are (at least) three different ways in which something can be missing from the map. Properly done, Babble can detect all of them.

i. Beacons

A beacon attracts attention. Like the summit of a mountain or the base of a valley, it has the feel of a local optimum. Circumambulation in known regions of the map spiral inevitably into beacons - recurring thoughts and preoccupations.

Beacons signal there is more to the story - the single marker on the map that you knew lazily as "procrastination" is actually a whole category of distinct mental phenomena. Once a beacon is identified, it needs to be blown up because there is too much information to be stored at a point. Imagine surgically expanding the beacon into a whole bubble of space.

Newcomb's problem is still a beacon for me. Every time I wander into its general region, I add a bit of detail to my understanding. I gain a little more respect for credible precommitment and iterated games and a little more certainty of one-boxing in superficially Newcomb-like problems in real life. Nevertheless, a beacon of residual unease remains.

ii. Holes

Holes in the map are gaps in your knowledge and models. You can infer the existence of simple and low-dimensional holes locally - they are usually marked by weak beacons. Knowing plenty about how baby humans develop, you infer that there is just as much detail to learn about crocodiles. Babble can sometimes lead you to fill in these simple holes out of curiosity, and find out that crocodile gender is determined by the temperature of the egg. Nevertheless, you don't have time to fill in all the little gaps.

Big gaps in your knowledge may be so big you don't even know they exist. If you walk in a straight line while staring at your feet, you'll miss most of the serious holes in the map. That's the value of circumambulation: detecting holes more globally when local detection is difficult or impossible.

The map may be a hollow sphere (or worse, a flat torus) that needs to be filled in. Walking around myopically, you think you live in Flatland. But if you Circumambulate mindfully, you might notice your Babble looks like this:

Detecting that three right angles don't make a triangle, you notice that your map has positive curvature, and there might be important insight at the center of this spherical hole you've been skirting around. Insight that you couldn't have found by walking in straight lines.

iii. Discontinuities

The confusion you notice as you Babble can signal holes in your knowledge, but it can also mean there are inaccuracies in what is already there. Figuring out what to do with a note of confusion is a delicate science. That slight pain in your chest - is it heartburn or lung cancer?

You make a circle around the concept, only to end up not where you started. Maybe it's a sign of a hole, or lack of precision. You think the map is a circle, but actually it's a covering space thereof, i.e. a much bigger circle. Your models are not sufficiently precise to distinguish subtle differences between points in the same fiber. It might also be the sign of a discontinuity, an internal contradiction in your map. Imagine the dismay of a cartographer who tries to build an atlas based on Columbus' reports of reaching India, unaware of the entire New World.

Discontinuities can be global just like holes, and require global solutions, and to rebuild models from scratch for a single discontinuity is usually too expensive. Identifying and eliminating these systematic biases - or at least correcting for them - is a central rationalist project.

2. Circumambulation as Battle Royale

Previously, I compared Babble to sampling from Google's PageRank algorithm: taking random walks on the Babble graph with random restarts. The randomness is biased by a weak, local heuristic.

But however useful PageRank is, it's only once in a blue moon that someone wants a global list of all extant web pages sorted by popularity. Google needs a search term and filters to be useful. Similarly, every Babble walk has to start at a basepoint, and it's less random than I suggested.

Narrowing down to a central truth is like a game of Battle Royale. Initially, many different hypotheses are spawned all over the map and fan out in pseudo-random walks. However, as Babble generates hypotheses and some of them are Pruned, we begin to triangulate the coordinates of the center. The safe area shrinks.

Babble is produced starting from a basepoint and restricted to a general region. As a clearer picture of the center is developed via Babble and Prune, the basepoint shifts closer and closer to it, while the region of interest shrinks like the safe area. These constrained random walks are forced to clash, and stronger hypotheses eliminate weaker ones or coalesce until at the very end, only one winner remains.

3. Focusing is Circumambulation

This recent post on focusing fits very cleanly into Babble and Prune. I recommend reading the whole post, but here are some relevant points:

The "big idea" of Focusing (according to me) is that parts of your subconscious System 1 are storing up massive amounts of accurate, useful information that your conscious System 2 isn't really able to access. There are things that you're aware of "on some level," data that you perceived but didn't consciously process (see blindsight as both concrete example and metaphor), competing goalsets that you've never explicitly articulated, and so on and so forth.
[...]
(There I was doing a super-fast scan over a whole bunch of possible words and phrases and explanations, all the while paying very close attention to my truth-detection module. I knew where the answer would be, and I knew its general shape, but I had to keep looking until I found something juuuuust right. It's like when you mentally stutter past five or six different comebacks to throw at your sibling until you find the one that's cutting, true, and okay-to-say-even-though-your-parents-are-listening.)
You get the idea. As the process continues, the picture grows more and more accurate, and evokes more and more of the underlying what's-really-going-on. I can feel a sort of click, or a release of pressure, or a deep rightness, once I say the thing that really completes the picture.

In our language, massive amounts of accurate, useful information are stored by System 1 behind the First Gate of Prune, and focusing is a particular exercise towards noticing previously subconscious information (beacons, holes, and discontinuities) and letting useful Babble through. Prune is a necessary part of this process, killing weak hypotheses and directing Babble in the proper area.

I have described a general, not entirely novel, approach to tending your mental garden, circling around the map via Babble in search of precision, completeness, and internal consistency. The upshot of this post is that Circumambulation is the right geometric imagery for this process, that geometric metaphors highlight the possibility of serious and nonlocal flaws in the map, and therefore that spiralling around the truth at length is rarely a waste of time.