Circumambulation

Link post

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.