28. Fractals, Foxes, and Hedgehogs 🔗

“The fox knows many things, the hedgehog knows one big thing” — Isaiah Berlin

A question: how do foxes and hedgehogs “know” fractal realities, self-similar across scales? How do they act on them?

Answer: badly but in different ways.

Cauliflowers collapse cognition: a thread

Fractals are the enemy of tractable ontology. They suck hedgehog attention down to lowest level/highest resolution phenomenology, destroying efficient abstraction hierarchies, and cause foxes to degenerate into vertigo-inducing scale-free apophenia. Why?

The only “efficient” kind of scale-free thinking is recursive thinking, and human brains are bad at it. Maybe we have limited stack capacity. This is why puzzles like the “blue forehead” are hard. The logic isn’t hard, but it involves recursion. discovery+ | Stream Real-Life TV Episodes

Imagine approximating a cauliflower with a solid that follows contours of largest florets using a parametrized mushroom-shaped primitive. That would cause both positive/negative errors (solid where empty, empty where solid). That’s a good example of a fractal map-territory error.

To get more accuracy you would use a smaller “mushroom” as your 3D tesselation primitive. All the way down to the smallest where you’d get “riced” cauliflower (down to rice-sized pieces). The basic idea of fractal models is to use geometrically similar primitives at all scales.

The “coastline of Norway” is a more conventional example but I prefer cauliflowers since they are things we act on, by chopping etc. The “right way” to chop a cauliflower, as my buddy @bumblebeelabs once observed, is “recursively”.

Recursive chopping is when you decide on a target floret size (say 1/2”) and break down big florets to roughly that size. It will never be perfect. But you can get most of the mass under a hard upper limit of size in the form of florets with integrity. There Will Be Crumbles.

Most good cauliflower recipes call for a floret-integrity-preserving deconstruction (I detest things like mashed cauliflower which destroy its character 🤬). But it’s possible to take a simpler chopping approach. Like the dicing of potato.

Chopping cauliflower with a potato technique leads to a lot of waste for a target size. Imagine “dicing” cauliflower into half-inch cubes using knife strokes that follow a cuboidal grid. You’d get unpredictable fragments. But the smaller your target size, the less it matters.

At the limit, to get to “riced” cauliflower, the technique kinda doesn’t matter: recursive chopping, dicing, and grating, all lead to roughly the same results: a cauliflower-insulting state good only for mushy potato or rice substitution. Structural destruction is path agnostic.

That’s what pure hedgehogs do: getting very detail-oriented by reducing the cauliflower to limit where fractal geometry gives way to the next lower level of organization (cellular). They destroy what they can’t grok, to reduce to a perfect “one big thing” lower-level idea.

Pure foxes otoh get attached to maintaining the integrity of the loose “floret” primitive to the point where they can’t recursively chop at all, since every cut, whether fractal or dicing, causes “non-floret-like error” to accumulate. There Will Be Crumbles. And stemmy bits.

Because there’s no “flat” (1 level/size band) approximate breakdown with no “errors” (ie only floret shapes and within a size band with non-trivial upper and lower limits), the fox is tempted to deep fry the whole, holistic cauliflower so to speak. Or more likely, do nothing.

(More technically, every reduction to a size band between 0 (riced) and 1 (whole cauliflower) either introduces new “fake” primitives like weird “stem junction cubes” that are artifacts of decomposition technique, or violates a lower size bound).

Because the fox “knows many things” (the florets in their analogical variety, across sizes and parametric shape variations), any chopping is too destructive and they freeze into inaction. This is one failure mode induced by fractal realities.

Because the hedgehog “knows one big thing” that won’t tolerate fractal variety, they destroy the whole thing and tame it with brute force. This is another failure mode induced by fractal realities.

Both are driven by fear of incompleteness of existing knowledge.

In both cases the solution is to accept that any action will change reality in a way that makes your previous knowledge incomplete. So you either have to destroy the reality entirely or accept that creative destructive action creates bits that don’t fit what you know.

Basically the only error-free map is the entire territory. Hedgehogs forcefully tame it into “rice”, foxes leave it untouched, representing the “floret gestalt”. Destroy the territory through reductionism, or treat it like a holistic map with no agency for you.

Non-fractal realities don’t have this property. For example, animals are not fractal. So a simpler level-by-level deconstruction works. This is Lao Tze’s butcher, taking reality apart at joints elegantly. Disassembly without fractal chopping. Our normal thinking is like this.

Under normal conditions, we tend to use “stack” thinking: a set of single-level, mutually exclusive, collectively exhaustive (MECE) mental models. We ignore fractal error. We move abstraction levels as necessary, trusting state to stay well-behaved.

Stack structures can be navigated with lightweight stacks in the computing sense. This is because they follow a finite ontogeny recipe.

But large-scale growing realities tend to be fractal in macro structure. When you try to navigate them with stack logic, things fall apart.

The @EpsilonTheory essay “As Above, So Below” in the “things fall apart” series gets at this. Epsilon Theory - Epsilon Theory

My gloss on that is: under normal conditions stack thinking in a fractal world causes fractal map-territory errors that self-correct via foxes and hedgehogs serving as checks and balances on each other. That’s complex systems homeostasis.

But in weird conditions, they compound fractally, collapsing at all levels. The widening-gyre effect. Hedgehogs become part of problem by getting destructive. Foxes give up in frozen inaction. The system gets ungovernable. Self-correction homeostasis breaks down, all unravels.

The solution is to “think entangled, act spooky” as I recently argued. To arrest and reverse a fractal collapse you have to think in fractal-native scale-free ways. How do you do that in practice? I don’t know yet. Working on it. Think Entangled, Act Spooky