Both and Neither (The H-Fractal)

I refer to the both-and-nether diagram fairly often. For me, it’s an idealized vision of the landscape of theories that can be developed through assumption switching.

Below is a 13th level diagram. The infiniteth level is a space-filling, self-similar fractal of my own invention, I think. [Oops, some research shows this is a well-known curve, the H-Fractal. I'm still quite sure I discovered it independently. It first appears in my notebooks around 1980 -- a little before I'd ever heard of fractals. Apparently, the shape has some significance in the design of microchips and micro-antennae. My application is quite different as explained below.]

I stopped at level 13 so that you can see the structure. I picture this thing as the space of all maps. Suppose that a line segment (of length x) represents a continuum between descriptors A and ~A. For example, look at the thick horizontal line in the middle of this image. As we slide from one end to the other a representation of yin becomes a representation of yang, good becomes evil, love becomes hate, levity becomes gravity. That is, each line segment represents a quality that has poles of difference. Now add a new dimension by imagining that each endpoint of that continuum is replaced by a new continuum at right angles to the first (with a length of x divided by the square root of two in order to keep the curve from running into itself). An interpretation will follow shortly. Continue in this way. That is, replace each successive endpoint with a shorter, perpendicular line segment. Here I've made longer lines thicker as well so one can better see the bones. This variable thickness messes up the space-filling aspect a bit. You can see the thickest lines are already making contact with the shortest ones. In the ideal representation, no line segment has any direct contact with any other segments except its immediate "parent" and two "offspring." The 2 to the n endpoints create a grid of dots that fills the rectangle more thoroughly for each new value of n. Two things to notice: 1) the paths from every endpoint to the center of the diagram all have exactly the same length (approaching (2 + sqrt(2))*x); 2) The graph is self-similar.

Both-and-neither diagram

In my specialized interpretation, each endpoint represents a different perspective from which the world can legitimately be described. Each line segment is a continuum or spectrum ranging from utterly x to utterly not-x. The world is both that understood from one such point or another but obviously not entirely from either. Both both-and-neither and neither both-nor-neither.

This diagram gives an illuminating way to pack many dimensions into just two. Because of the unique scaling, each nonintersecting perpendicular of a particular length can stand for a different variable or idea. One gains a tremendous amount of simplicity and comprehensibility -- just try to picture 13 dimensions in some other way -- but of course much is lost as well. For example, the represented 13D plane doesn't have what mathematicians call a metric. We'd like nearness to indicate similarity; we'd like distance to have an interpretation. But here two very near points can be parts of two very distant branchings. Perhaps there is an interpretation that preserves a sort of mysterious metric, but it isn't clear to me how one would go about it. Maybe there's a discovery to be had there. I take it back. Nearness along the curve does have meaning; it's only when you venture from one point on the curve to another through white space that the metric is completely lost. Since ultimately there is no white space maybe this isn't a problem.

This may at first seem to be a stretch to my reader, but for me, this diagram is a nice representation of the map-territory relationship. Each endpoint/segment can stand for a particular map, a particular and definite way of describing the world (the LRRLLLRLRRLL perspective), and the collection of them for all the switched and/or hedged versions (see elsewhere) of the map. The territory is described by all of the maps but by none of them very well. Each is simultaneously an overstatement and an understatement. Each connected pair sets up a both-and-neither situation. Both yin and yang and neither yin nor yang. A pair of connected pairs gives a "both both-and-neither and neither both-nor-neither". Ad nauseam. In toto, the collection of maps gives a super map that represents the territory better -- if only we could understand it with our feeble seven-plus-or-minus-two minds. Martians may be better at seeing the point.

The image resonates for me, but interpreting the diagram in this way can be tricky to instantiate. I'll settle here for a slightly different and more mundane interpretation.

Let me describe a super map of, for example, political perspectives using this imagery. I'll only go to a 3rd level diagram with eight endpoints (see below) to keep things relatively manageable. I'll use right and left in their political senses. (More commonly I have yin on the left and yang on the right.) The right end might be small-and-decentralized government and the left big-and-centralized government. So this first dimension is about someone's preferred degree of government involvement or control. A second dimension might involve something about the emphasis on individual good (right) versus collective good (left).

The small government preference/perspective could thus split off into small government with a collective sense of the good (tribalism/communalism) above and small-individual (libertarian/free-market-capitalism) below. Likewise, big government splits toward socialism above and liberal democracy below.

Another dimension might involve secularism (left) versus sacredness (right). With a little imagination, one can picture libertarians splitting into those who, on the one hand, find that regulation is inefficient and stifles the production of wealth and, on the other hand, those who object to governmental intrusions in their private lives and beliefs. That's one of four splits at this level. How might communalism split? Efficiency and profit through cooperation and teamwork rather than competition (the old Japanese business model) versus the communalism of the Amish, serving God and the good through group effort. Two down. There are secular and spiritual versions of socialism -- Sweden versus the Quakers maybe? The so-called East Coast media liberal elites are a sort of secular big government individualism. Who will exemplify the eighth and final node of my diagram sacred-big-individual? Maybe militaristic interventionist nationalists. If we were to go on, a fourth dimension might be freedom (right) vs. rights (left) and the eight nodes would have to become 16.

Let me summarize the examples above:

1) Small-government collectivist secularism [(RLL) Japanese business model]

2) Small-government collectivist spiritualism[(RLR) Amish, tribe]

3) Small-government individualist secularism [(RRL) free marketeers]

4) Small-government individualist spiritualism [(RRR) the Freemen]

5) Large-government collectivist secularism [(LLL) Sweden]

6) Large-government collectivist spiritualism [(LLR) Quakers]

7) Large-government individualist secularism [(LRL) upper middle class liberal Democrats]

8) Latge-government individualist spiritualism [(LRR) Americanism spreaders like Dick Cheney]

This effort is a little lame, but I hope the point of the exercise is clear enough. The diagram embodies multiple perspectives in a rational way that reminds us of the relativity of each perspective while also reminding us of the monolithic territory (white space) on which is all rests. For any one place, there's only one way to organize a government.

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All Possible Maps

Over the last 50 years, I've spent an awful lot of time daydreaming about it, so I want finally to try to develop a picture of the space of all possible maps -- ones that show insight, accuracy, connection, etc. to the territory.

This effort is bound to be connected to the Allegory of the Cafe, the implicate order, Joelesque taoism, and especially the both-and-neither diagram. My ultimate inspiration might be my teenage whimsy I called the Melonquescence -- the set of all things that have yet to be imagined -- which itself must have been inspired by brief exposure to Russell's paradoxical set of all sets that aren't members of themselves.

First, a few premises.

•The map is not the territory. Duh.

•A nascent map of certain aspects of the territory can be developed from assumptions in a somewhat Euclidean manner. That is, they grow out of the assumptions 1) by deductions from those first principles, but also 2) by inductions based on evidence (which is itself interpreted through the lens of the assumptions). Change the assumptions and change the maps that develop, obviously. Furthermore,

•There are no correct assumptions. Only a God could have made any set of assumptions correct. There is no way to hold up a particular assumption to evidence and decide if it's a good one. Occam's Razor is ultimately irrelevant to truth -- it just helps we poor human thinkers deal with the immensity of the task. Thus, statements that seem patently false can be legitimate assumptions. In that case, the complement of other assumptions might be built around why the first assumption isn't really false.

•Thus, we can come up with a reasonable sort of set of assumptions and create a galaxy of maps to cover the gamut of observations (via deduction, invention, reverse engineering). Maybe the totality of Euclidean geometry would make a good example of a galaxy in this sense. There is a huge and perpetually growing collection of theorems derived from Euclid's axioms and the "laws of logic." These theorems stretch tentacles into all sorts of geometrical realities.

Such a map galaxy exists within a universe of many, many such galaxies such as various non-Euclidean geometries. The universe of map galaxies is a rational set of points (metaphorically at least) that approach the irrational (or real) points of the territory (which can't itself be legitimately thought of as a space since IT lacks essential mapness)

I am particularly interested in

•a good process for generating sets of assumptions that lead to map galaxies. What I call Assumption Switching is one such process. Various yin-for-yang switches from say standard scientific assumptions lead to galaxies intimately tied (orthogonal maybe) to those standard galaxies

•the structure of the relationship between galaxies. The relationships are a big part of the ultimate meta-map; a claim which I'll have to justify at some point.

Example of a standard assumption, informally expressed: Things change only when caused to do so. The world is implicitly reluctant and inert and must be coaxed or coerced to act, so we are interested mainly in explaining the circumstances that lead to change. In the simplest and oldest interpretation, change manifests only through force (i.e. coercion).

Switch from that assumption: There are many, many to choose from. The most extremely antithetical to the former assumption, and thus the one I have the most fun with is "Everything is trying to happen at once." Rather than being inert, the world is intensely alive, fecund, and creative. Talk about the set of all sets! Now we will be interested in explaining how some things sometimes remain unchanged, how anti-jump muscles flex. There's no use asking this theory to explain what mechanism makes everything try to happen at once -- it's an assumption.

I hope that the ridiculousness of this new assumption inspires you to see the equal ridiculousness of the former. Neither would seem to represent how things actually behave. The observable world is predominantly neither about change nor stasis. I like to say that the territory is egalitarian. As in any interesting case, both A and the opposite of A are embodied in the world. Our tendency -- maybe even our psychological or linguistic need -- is to pick one side of the debate as natural or original and thus not to be explained, and the other side as unnatural or derivative and in need of explanation. Logic moves in a line from A to B but the world the logic illuminates does not. A and B were always there from the start. Logical deductions perform miracles, but they ultimately produce phony naratives.

Let me briefly address "Everything is trying to happen at once" so that it might sound a little less absurd. If there's an inherent impulse to move object X to the left, there's also another impulse to move object X to the right. One can imagine that the impulses cancel each other out. Imagine a circle of electric fans pointing toward the center of a circle occupied by a beach ball. The ball is stuck there, maybe jiggling a bit in the turbulent airflow -- like quantum indeterminacy or Brownian motion. Now turn off one fan. The imbalance leads to movement toward the turned off fan -- voila, change! Likewise, the seething undercurrents of contradictory sociopolitical wills in a nation (fans standing for fanatics) tend to cancel each other out peacefully on the beach ball of state until one or more of the factions stops blowing or another one starts blowing harder when radical changes begin to happen. Maybe vacuum cleaners are more apt than fans. By this view, time/history is the slow leaking of events allowed by asymmetric canceling impulses. Almost everything gets canceled out by the saturation of fans blowing in different directions. There are lots of situations that do seem to depend on imperfect cancellation of myriad tendencies -- through, for example, the sum of probability amplitudes in quantum mechanics. Stasis and change are statistical -- probabilities tending toward certitude through large numbers.

Well, as in most of my writing, so far there's been nothing but tangents all the way up! I promised to develop a picture of the space of all maps. Have I even begun to do that? [Don't beat yourself up, Joel. You're good enough and strong enough, and, gosh darn it, you have important things to say. If nothing else, you're exhibiting everything-trying-to-happen-at-once in your head! Narrative linearity is overrated. This is philosophical poetry, popping out little jabs of insight that cannot be sustained. Does that diminish the value of the insight. Well, yeah, but not entirely.] I guess I'll take whatever moments of lucidity I can muster. Okay, I can proceed now...

Okay, we have these assumption-driven galaxies. There are a countable infinity of such galaxies, even if deterministic science wants us to believe that only one of them is correct. Does this even begin to cover all of the points in the space of all maps. Doubtful. I just compared the infinite list to the rational numbers that come plenty close to touching all the points but only account for a set of measure zero. That might be about right. Again I am thinking of my both-and-neither diagram which is an H-fractal in a 1 x 1.414 rectangle. That rectangular area is the set of all maps. and the H-fractal extends its tentacles deeper and deeper, but only the unrealized limit extends to all the "irrational" points. Each path from the center moving outward taking a series of increasingly frequent right and left turns represents a galaxy. each turn is a binary choice of yin or yang orientation on a single (axiomatic) question (like "Does the preferred governmental policy-making structure involve few or many people?" or "Does one's ability distinguish the weights of two handheld objects primarily involve absolute or relative differences?". That is, questions about things that have endlessly debatable consequences.)

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A few other observations:

1) As here, analysis leads to splitting, which leads to subtlety and novel juxtapositions, which with luck leads to insight. We like things in black and white, but the more you analyze, the grayer it gets, the more strange the bedfellows.

2) I want also to point at the relationship between reality and our descriptors for reality (like left and right). Ultimately, this relationship which we assume to be sound and stable is kind of arbitrary and bound to collapse.

My politics are both right and left, and neither right nor left. Both both-and-neither and neither both-nor-neither.

It's very hard for me to go on to a fourth dimension with 16 examples. That's the law of 7 plus or minus 2 again (see elsewhere). People just aren't designed to think that minutely.

I've done other "octets" like this. My dichotomy is more likely to be yin-and-yang or love-and-will than right-and-left, but it doesn't really matter. Here's one I'll be making up on the fly.

dimension 1 LR: The "naturalness" of change versus stasis. Isolated coercive forces vs. cancellation of persistent universal tendencies

dimension 2 RL: determinism vs. indeterminacy. The pre-existence of the future vs. the unfolding of novelty

dimension 3 LR: consciousness is fundamental vs. consciousness is epiphenomenal

RRR Stasis-determinism-epi : "hard" Newtonian mechanistic science of universal mathematical laws and thus strong AI

RRL Stasis-det-consc: Cartesian dualism?

RLR Stasis-indet-epi: Quantum reality?

RLL Stasis-indet-consc: Western mysticism?

too hard after this.