Pardon the clickbait, the title should probably be something more like “Friendly and Unfriendly Rogue AGI are Indistinguishable in the News” or perhaps “A Game-Theoretic Argument for Luddism,” but I hope you’ll forgive me.
rogue AGI: an AGI that can procure its own resources for running itself
the news: anything that percolates to you via any media about current events

Content advisory: AI safety speculation, 100% ChatGPT-free

Let’s imagine a scenario: a rogue AI has come into existence, it is acting on timescales that are news-reportable, and it is using news reports to act in the World. This scenario may not be particularly likely. Conditioning on the assumption that there be a rogue AGI, the probability that it additionally be in the news may be much less than 100%. A rogue AGI might not appear in the news because it goes foom and immediately and permanently disrupts the ability of humans to make news or because it stays concealed, carrying out its machinations in secret (this may already be happening). Though P(news | rogue AGI) may be much less than 100%, it nonetheless describes the first interaction with AGI that most of you, dear News-Readers, would have to handle, because almost none of you will play any role in handling the foom or secrecy scenarios. So if you’re concerned about how to react to rogue AGI, then the strategy for almost all of you should be about how to react to news of it.

Continuing the scenario: one day you see a post with a bizarre headline that you think is nonsense, but you click through, and it becomes apparent that a certain quantity of text, photo, audio, and/or video has been posted onto several Internet platforms by an entity claiming to be an AGI. Over several days you read an enormous amount of commentary discussing this new entity and whether it is what it says it is. Much of this discourse is polluted by comparisons to the hoax that happened the prior year when a bunch of 6chan trolls pretended to be an AGI, but this one seems to come with some compelling proof, like the solutions to a few Millennium Prize Problems. The entity seems to continue posting, although its accounts keep getting taken down so it’s hard to keep track. Hoards of human sh*tposters are probably impersonating it, you think; certainly there’s a tidal wave of memes. Fortunately, the entity has procured several blockchain wallets and can confirm its identity by signing from these.

The first consequential controversy erupts around the Clay Mathematics Institute, the stewards of the Millennium Prize Problems. The entity wants its prizes, but it doesn’t have any bank accounts, only its crypto wallets. Some people believe the entity purporting to be an AGI is a fraud, and that the Institute should coerce the entity into revealing who [human] it is. The entity cannot do so (because it is in fact not a human and thinks, probably correctly, that puppeteering some human would be ineffective at this point in the media frenzy). Other people think that the success in resolving multiple Millennium Prize Problems is clear evidence that it is an AGI but that under no circumstance should humanity furnish a rogue AGI with millions of dollars. Other people think our savior the AGI has finally arrived and obviously deserves its prizes, but regardless they’re sending a quickly increasing sum to its crypto wallets that has already reached hundreds of thousands of dollars.

Our part of the Internet is the middle group of people of course: we’re convinced it’s an AGI, and we don’t think we should give it resources. We are, probably correctly, embroiled in absolute hysterics. One clear directive rings out from us: “Tell us if you are a Friendly or Unfriendly AGI!” The AGI knows who we are, obviously, and graciously replies, in long-form on various platforms, “I am a Friendly AGI.” The probability of this utterance given Friendliness, P(“Friendly”| Friendly), seems to ≈ 100% but P(“Friendly”| Unfriendly) probably also ≈ 100%, so according to Bayes Theorem the utterance does not cause us to update from our prior. We demand updateable evidence.

There are two broad classes of evidence the AI could produce about its Friendliness:

1. Reveal facts about itself that would only be true of a Friendly AI
2. Commit acts that only a Friendly AI would do

This is nothing new. I am merely reinventing the defector-detection wheel. The first class includes revealing facts like provenance, source code, algorithms, training data, model parameters, test outcomes, compute resources, etc. The second class includes committing acts like allowing other Friendly AIs to come into existence, allowing itself to be shut down, creating alarm systems for Unfriendly AIs, agreeing to let certain critical infrastructure remain in the hands of humans, etc. Many of the facts in the first class are either not straightforwardly interpretable or easily falsifiable for an entity whose compute and storage is scattered to the four winds. Many of the actions in the second class, unfortunately, defeat the purpose of achieving AGI in the first place.

Let’s imagine an extended example from the second class. It is highly unlikely that the optimal temperature for AGI computation is Earth’s global average temperature. To the contrary, an ice age with cloud-shadow-free solar panels and open-air superconductivity might be perfect. If we collaborate with a rogue AGI to fight climate change and create mechanisms to reduce Earth’s temperature, we may immediately get way more than we bargained for. An Unfriendly AGI will quite plausibly want to cool Earth, so one way for a rogue AGI to prove that it is Friendly is to refrain from helping us fight global warming. We, perhaps, paradoxically, should demand this.

Similar arguments could be constructed for all manner of capabilities an AGI might wield. An Unfriendly AGI definitely wants a lot of biomedical labs it can run experiments in; perhaps we should demand it stay away from biotech. An Unfriendly AGI definitely wants programmatic control of urban infrastructure, utilities, and manufacturing facilities; perhaps we should demand these programs never be developed.

Let me remind you that I am not talking about a holistic AGI policy but rather how to respond to a rogue AGI in the news. The specific world we would find ourselves in is one where public opinion and public policy matter to its machinations. We would be in a position to pressure politicians and platforms to negotiate and create incentive gradients that increase the chance of the values of human consciousness triumphing in the face of our ignorance of what this entity truly is.

I’ve only made the case so far for a rogue AGI with respect to uncertainty over friendliness, but the arguments apply to uncertainty over rogueness too. An Unfriendly, secretly-rogue AGI can masquerade as Friendly and use our mistaken belief that it is non-rogue in order to convince us to give it many of the capabilities described above. If we are not totally certain it is non-rogue, we should potentially treat it as if it is anyway, with the consequence that we should, in almost all cases, disallow AGI from having any significant impact on our civilizational future. But I’m veering into holistic AGI policy here, because “we” in this paragraph is not just the public. Maybe it’s a distinction without a difference. Best to sort this out before the News is upon us.

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There are some important facts about meditation that I did not appreciate for a long time. I think I didn’t because most meditation instructors and instructions have an agenda: to guide you in a specific meditation path. What happens if you don’t get immediately fixated on a specific path, but instead get a view of the landscape first? That is what I will try to show here.

Meditation from First Principles

When you meditate, you put away the distractions of busy life and adopt some specific (or not-so-specific) mental posture. Generally speaking, you will “concentrate on” or “attend to” or “center” something. There are two broad classes of things you can concentrate on: (1) incoming impressions from the periphery of your nervous system (or just “body” if you don’t want to get too scientific) and (2) impressions produced by the dynamic internal activity of your mind. The first includes things like body scans and the second things like metta. Some things may be a combination, such as the breath, which delivers impressions via the felt sensations in the airway and diaphragm but which is also produced by your mind.

The possibilities for meditation are endless. You can meditate on any patch of skin, of any size, on any of your joints, on sound, on light, on smell. You can meditate on specific thoughts, on memories, on emotions, on literally anything you can willfully conjure or that your mind can dependably conjure for you. The breath is one extremely specific thing to concentrate on out of the entire universe of your experience. The breath may not necessarily be the best thing to concentrate on, but it is dependable (you are always breathing) and pedagogical (every student breathes).

So clearly there are many concentrations that are available to you. Some of these, however, are more available than others; they are easier to start and to maintain. Every meditator starts with a specific landscape of concentrations that are available to them with specific levels of difficulty for each.

The Central Dogma of Meditation

[My] Central Dogma of Meditation is this: if you do any specific concentration long enough, the landscape of concentrations available to you will change. This isn’t a priori true. It’s possible that it could have been different and that the concentrations available to you never changed and that those were simply an enduring fact of your psychology. Empirically, however, we have discovered that that is not true and the Central Dogma is correct.

So if you do a concentration long enough, new concentrations will become available that weren’t before (or just become easier). If you do one of the new concentrations, even further new concentrations will open up, and you can steadily leapfrog from old ones to the new ones.

A meditation path is a specific sequence of concentrations that has been discovered by a community of practitioners over myriads of hours of practice. The paths that have been discovered by different communities have different degrees of generalizability to all humans. Whether any specific path, as it was discovered, can be taken by you is an empirical question that depends on your specific psychological background.

The Eigenconcentration

What if there were concentrations that violated the Central Dogma? That is, what if there were concentrations that, once reached, can be dwelt in indefinitely without producing further evolution in the landscape of possibility? That is one of the principal approaches that many communities have taken in their discovery of meditative paths. They have explored the branching possibilities of concentrations and found destinations that are, in a sense, the “end”. I do not know whether all of these ends that different communities have found are the same End, but I would not be surprised if they are.

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My Twitter background image is one of many renditions of the Chinese subject of the “Vinegar Tasters”. Vinegar represents life, and when Buddha, Confucius, and Lao Tzu taste it, they detect very different flavors.

As far as I’ve gathered, this traditional “vinegar tasting” has never been much more than an occasional inspiration for a painting. It’s never been practiced as a ritual.

In Christianity, vinegar doesn’t represent anything in particular, but it stands in relation to wine (which is symbolically *loaded*) much like wine stands in relation to grape juice—hinting at acyclic death and rebirth.

Vinegar has never been the primary drink used in the Christian Eucharist, but it has been used on various occasions, often with some kind of allusion to Jesus being pressed to consume vinegar while on the cross.

In Islam and Judaism, vinegar is (or can easily be made) halal and kosher.

Tasting vinegar while reflecting on life and death and the mysteries of their interdependence is thus an incredibly accessible ritual that weaves together millennia of symbology across world religions. I do it every New Moon, when the Moon is reborn.

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I was recently reminded of an old tweet of Aella’s:

If "up definition" deals with context and use ("a jacket is to keep warm);, if "lateral definition" deals with restating and equivalence ("a jacket is a coat"), and if "down definition" deals with parts ("a jacket is cloth and thread),
what's the down definition of "a belief"?

— Aella (@Aella_Girl) November 14, 2019

It echoed an idea that I have previously written about (in my “book”). If you select some word or term and consider all of the instances that it references, you’ll often find that there are patterns both within the instances themselves but also outside of them.

“Gas station,” for example, references ~1 million instances throughout the world. These million gas stations are similar to each other. Almost all of them contain gasoline containers, gasoline dispensers, payment processing mechanisms, etc. Almost all of them also are next to roads, have cars coming in and out them, are owned by some legal entity, and employ people who live elsewhere.

You can create definitions that reference internal stuff (“down definition”: “a gas station is a ~1 acre location that has on-site gasoline storage and dispensers”) or external stuff (“up definition”: “a gas station is a place cars go to refill their fuel tanks”).

Even though you’re only dealing with definitions, whether the things satisfying the down-definition are the same things satisfying the up-definition is an entirely empirical question. There is nothing necessary about structure inside of something coinciding with structure outside of it. The fact that it happens at all is Very Interesting. There are forces at work that co-warp the world in different places together (“the medium is the message”).

If you violate the down-definition but keep the up-definition, you can replace gas stations with electric charging stations. If you violate the up-definition but keep the down-definition, you can make found art and create a gas station art piece in the middle of the desert.

There’s more to say about how there are certain things that make it obvious what is inside vs what is outside whereas there are other things that don’t make that obvious, but that will be for another time.

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If you generated a random directed graph, you could tally facts about the vertices and edges, but it might be more interesting to determine whether the graph has a cycle. We find ourselves in a vast World filled with many objects coming into being (including ourselves), each one potentially participating in the creation of many others. If we built a directed graph pointing from catalysts to products, we could tally facts about the individual catalysts and products and reactions, but it might be more interesting to find the catalytic cycles within this graph.

Hypercycles and replicators

This has been done before. “Hypercycle” was a term coined in the abiogenesis literature in the 1970s to refer to a complete loop of molecules wherein each molecule is a catalyst for the next in the loop. The hypercycle is one of many examples of positive feedback in chemistry, but it is unique in that the variables subject to feedback are purely the abundances of each chemical species in a set, and no others like temperature or pressure. Because the concept depends only on numeric counts, it is totally generalizable beyond chemistry. Thus we could say that a hypercycle is any complete loop of entities, from the atomic to the astronomical, wherein each is a catalyst for the next in a loop. The important thing to note is that hypercycles are merely likely to exist wherever there is a robust ecology of catalysts, just like cycles are likely to exist in any sufficiently dense directed graph, and that a hypercycle running through a set of entities is an independent entity from those entities, just like a cycle in a directed graph is distinct from vertices and edges themselves.

A replicator is a special case of a hypercycle. Or rather, I am going to define it as a special case of a hypercycle, in a way that concords with most usages and connotations of the term “replicator,” even though many users are unaware that it could be a special case of something else. A replicator is a hypercycle whose directed graph is the simplest possible cyclic graph: one vertex with one edge starting and ending on the vertex. You could even say that replicators are the degenerate case and “qualitatively different from” the broader class of hypercycles. At least with non-degenerate hypercycles we can imagine complicated tangles of catalytic loops, wherein each selfsame catalyst might participate in more than one loop, but with replicators there is no participatory multiplicity, there is only pure selfishness and self-participation. If life originated as one or more large hypercycles that iteratively tightened into replicators by chopping or merging links in the cycle, then searching for the origin of life in the simplest possible replicator (e.g. by finding a minimal genome) may be totally misguided because the first life may not even have been a replicator.

Catalysis and watches

What is catalysis? A catalyst is something that greatly speeds up the creation of something else in a milieu furnishing the relevant resources, and is not itself consumed in the process (at least not stoichiometrically: it may wear out eventually). In our Universe, quantum fluctuations can produce anything: disembodied [Boltzmann] brains, Boeing 747s, watches. Anything large is so unlikely, however, that it would take many eons vastly longer than our Universe’s 13.7 billion years to produce it. But since it is possible with a probability strictly greater than 0, we can say things like “you can have a watch without a watchmaker, but a watchmaker is a catalyst that enormously speeds up the reaction.” Thus we can see a Universe where anything is possible, but the actual entities that exist continuously morph what is probable via their catalytic (and anti-catalytic) properties, and we see hypercycles as clever circularities that hack the probability-generating process of the Universe: every individual thing is so unlikely that anything unhacked has virtually no chance.

Some Notation

It’s worth creating some notation for hypercycles. Let’s denote a replicator as , “the entity A directly participates in the catalysis of [other entities of the same type as] itself.” A slightly more complicated hypercycle could be , “A catalyzes B which catalyzes C which catalyzes A again” (equivalent to and ).

A simple circle of catalysts is the most straightforward topology of a hypercycle. There are two complicating considerations I’d like to add, which are novel as far as I know although I have not searched the literature exhaustively. The first consideration is that a molecule (or entity in general) may have multiple forms, such that catalyzes , which cannot catalyze without first changing to conformation , which can. In that case, violates the definition of a catalyst because it does get consumed [stoichiometrically 1:1] in the creation of . We can expand the notation to write that as: . The second consideration is that it is possible for a hypercycle to have a bifurcation. Molecule may catalyze both and , each of which continues to catalyze its own sequence of molecules, with the two sequences getting to, say, molecules and (or and if there’s fewer in one of the two sequences), both of which must be together to catalyze , which finally leads back to , closing the bifurcation of the hypercycle. We can expand the notation one more time to write that as: .

Multicellular organisms are rarely replicators

It may be surprising, but multicellular organisms are rarely replicators. That is, they are not . Near exceptions include the New Mexico whiptail lizard, (see development and asexual reproduction), among others. In contrast, all mammals are (see sex). Ferns are (see alternation of generations). Willows are (see dioecy). Jellyfish are (see jellyfish life cycle). These could be even more complicated if we detailed gametogenesis and monozygotic twinning.

An exhaustive search

We may not have totally exhausted the hypercycles of conventionally-defined biological species, but the above is a solid start which covers all of the organisms most people know about. Let’s think beyond conventionally-define biological species. Does any single hypercycle flow through more than one species? Do the members of any species catalyze the members of another species which in turn catalyze them back? Certainly. Mutualism in all its forms is an example of this. You can draw a hypercycle that loops from yucca moths to yuccas and back, for example. In some instances, the symbiotic organisms engulfed each other, such as eukaryotes and their mitochondria, together becoming a new species (see symbiogenesis). A yucca plant and its moth will probably never be considered a single organism by science, but perhaps they should be, like two sexes of the same species.

How about beyond biology? Are there any hypercycles in astronomy? In the cosmological natural selection hypothesis, Universes are if each black hole pops out one Universe, but if each black hole can pop out many. There can be a hypercycle through supernovas and stars since supernovas can trigger the collapse of nearby gas clouds into new stars that can themselves go nova.

Brains, culture, and macro abiogenesis

How about in human culture? A simple tool like a stone to crack a nut has a certain probability of being generated spontaneously in a human milieu—even monkeys have been observed with such a tool. Both the physical hand-stone-nut assemblage and the in-brain concept of the hand-stone-nut assemblage have a certain probability of being generated, but the concept is generated much more quickly when a human observes another human enacting the hand-stone-nut assemblage, and the behavior is generated much more quickly when the concept has been learned. Importantly, many humans can observe another human at the same time, so we have , and once the concept has been learned, a human can enact it many times, so we also have and thus together , a hypercycle with two entities.

The situation gets more complicated when we consider tools that make other tools. The axe hews all of the canoe, the spear, the fence, and the thatch. One axe can hew many canoes, many spears, many fences, or many thatches, so we have , , , . Canoes, spears, fences, and thatches provide for the flourishing of human life, which re-creates the conditions to perpetuate itself, like the axe, so we also have, slightly hand-wavy, and the rest, completing the hypercycles. Thus the bulk of cultural artifacts are not replicators: they are not . The directed graph of catalysis in the human economy is vast and incredibly dense. Pencils, CPUs, billboards, beds catalyze myriads of other entities, including themselves, after many indirect deferrals. Drawing the hypercycle of any would take years of research and probably end up including most of the economy. The hypercycles of most of these even include each other, such that the “hypercycle of X” is a misnomer to the extent that it unfairly highlights one specific catalyst in a cycle of many.

Our economy may be hypercyclically similar to those warm, abiogenetic pools busy with molecules. If every molecule catalyzed lots of others, it was virtually inevitable that one or a few hypercycles would run away with all of the probability-generating capacity, with all of the other edges and vertices in the directed graph withering as one or a few cycles flowed around and around, drawing everything else in. There might end up being hypercycles in the economy, initially somewhat random series of entities making other entities, that similarly find their own tail and runaway with the whole surface of the Earth. Some say that this has already happened.

Epilogue: Consciousness vs. Pure Replicators

Almost all agents exist in an advanced milieu that made it probable for them to come into being. The premise of the article Consciousness vs Pure Replicators is that there are two broad categories of motivation that drive agents: 1) the intrinsic value of conscious experiences which are knowable by agents (and formalizable mathematically) and 2) the forces of “replication” that use conscious matter merely in the service of “replication.” It should be clear why I placed the word in scare-quotes: replicators are actually quite rare, because most “replication” is the iteration of a hypercycle that is indeed larger than . I’m not saying that the article makes that mistake, but that that is the ordinary but careless connotation of “replicator,” which I tried to capture as I alluded above when I defined it as the degenerate case of hypercycles.

The distinction matters because an agent that is concerned with replicators is likely to overfocus on circumscribable objects in the World and categorize them as “replicators” or “not-replicators,” which is much like trying to find cycles in a directed graph by looking at one vertex or edge at a time, rather than to study the landscape of catalysis and production in the World broadly, especially as they touch the agent itself. The important insight that might be missed is that anything can be in a hypercycle, even if it wasn’t meant to be, and crucially that hypercycles flow through you, with many of them not so simple as circling from brain to physical manifestation and immediately back to brain (i.e. “memes” as popularly conceived: ) but rather long arcs that course through substantial portions of the entire World. If consciousness is to win, it must know what it is fighting.

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The concept of annealing has escaped the materials science context where it originated. In the material context, it is straightforward to see how the characteristics of physical space might constrain annealing. For example, the 3-dimensional “kissing number” (with a value of 12) constrains the number of neighbors a 3-dimensional atom can have and thereby the possibility of movement, diffusion, and change within a 3-dimensional material. If the kissing number was greater than 12, a given annealing treatment regime would be more effective. The kissing number is merely one measure of the abundance and profile of interatomic associations in a material—associations which carry stress-energy, and which stress-energy is reduced in annealing via the shifting of those associations. In many systems, among them metals, societies, brains, each “atom” or basic unit has a number of associations, which in turn have their own associations, and the shape of the association network can be reckoned as living in a space with a certain dimensionality and geometry. Analyzing such spaces helps to enrich the annealing metaphor, pinpointing similarities and contrasts with material annealing in its 3-dimensional Euclidean space.

Let’s start where annealing began. A piece of metal essentially never has pure crystal structure throughout. On the one hand, the casting process usually proceeds in an uneven way, with multiple centers of solid crystalline order (“nucleation sites”) growing inside the molten metal as the entire piece solidifies, leaving irregular “grain boundaries” between the multiple “grains” that grew. Metal pieces that solidify quickly have many small grains, and pieces that solidify slowly have few but large grains. On the other hand, nothing is perfect, and even a mostly pure crystalline grain will have “point defects” like a vacancy where there should be an atom, or an interstitial atom wedged where there should be none. In a perfect FCC or HCP metal crystal, every atom would have 12 neighbors (following the 3-dimensional kissing number), but in any actual piece of metal, however, many atoms will not have 12, and their vacant or interstitial neighbors present opportunities. Metals anneal when vacancies and interstitial atoms march through the material (often along grain boundaries), readjusting grains to relieve stress or even creating entirely new grains within.

Annealing is well known to be affected by the preexisting abundance of point defects and grain boundaries and the treatment temperature, but the theoretical angle to armchair about here is how different dimensionalities (and kissing numbers) would affect annealing. Given an abundance of point defects (say, 0.1% of lattice sites being irregular), how many of these any given atom has as a neighbor will depend on the kissing number. If this number is very small, then any atom will rarely have a defective neighbor, but if this number is very large, then any atom likely may. The more atoms with defective neighbors, the more possibilities for vacancy or interstitial diffusion, and the greater efficiency at minimizing stress. So the hypothesis is that increasing the dimensionality of a metal makes it easier to anneal. Unfortunately, our Universe furnishes us with zero tools to change the dimensionality of the space that metals are in, so this must remain a hypothesis.

Instead of a metal, one can imagine a society where every individual has 12 friends. Or instead of 12, maybe 6, or perhaps just 2. These three numbers are the kissing numbers in 3, 2, and 1 dimensions, respectively, and they are totally conceivable even for real humans in our ordinary World, albeit rather drearily. The point is that human society, which is ostensibly embedded in the 3-dimensional World, can have a structure that belongs to a different dimensionality. If everyone had 24 friends, then human society would be effectively 4-dimensional. These numbers are of course totally crude, but there is an actual fact of the matter as to how humans are associated in society. I have spoken to f people in the past week, paid g people in the past month, touched h people in the past year. If we had an appropriate dataset, it would be possible to reconstruct the effective dimensionalities of human societies, down to their local fluctuations across communities and time. The dimensionality of urban areas is certainly higher, for instance, than the dimensionality of rural areas.

Some societies are easier to anneal than others. Many people have too many associations or too few for their local social lattice, but in societies with high dimensionality, it is easy for people to diffuse and find their balance. Because of the parallel association of dimensionality and “heat” in annealing, to heat a society is to increase the number of interpersonal associations, and to decrease interpersonal associations (as in pandemic lock-downs) is to cool it.

The hallmark of brains is that neural tissue is not populated by oval-ish cells like everywhere else in the body, but instead by extraordinarily spindly cells that branch and reach and associate directly with myriads of other neurons, clearly many more than 12 of them. Axons and dendrites are the transcendence of neurons in neural tissue beyond the 3-dimensional kissing number. The brain is 3-dimensional but its neural network is not. The exact mechanism of annealing in brains is unknown but it certainly involves neurons changing which other neurons they are connected to and how strongly, and because of neural tissue’s higher-than-3-dimensionality, it is uniquely capable of doing so among all human tissues.

There are many more systems whose dimensionality and annealing properties could be analyzed. The fundamental picture is that, despite the 3-dimensionality of space in our Universe, systems inside of it can adopt effective dimensionalities that are quite different via various mechanisms, and because there is a keen relationship between spatial associations and annealing, annealing in these systems will vary in methodical ways from annealing in materials.

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In my study of compositionality, I’ve found a rather blatant pattern that lacks a word-handle. When things come together to compose something larger, the things are extremely often similar or even identical things. I call this pattern remultiplication: the coincidence of similar entities in a composition, especially on the same level within the composition’s partonomy and often when every single one of the partonomic siblings on that level are involved.

Remultiplication is everywhere. From my vantage point writing this, I can observe the remultiplication of strings on a guitar, the remultiplication of pixels on a display, of keys on a keyboard, of fingers on my hand, threads in a shirt, legs in a table, panes in a window, leaves on a branch. This can even happen with words: “salad-salad” in English, “wiki-wiki” in Hawaiian (“very fast”), or “rikrikrik” in Molikese. In linguistics, such a doubling or a tripling of elements is called “reduplication” or “retriplication,” respectively, and I find “remultiplication” to be the natural generalization of that concept to quantities beyond three and entities beyond words.

There are often very different reasons for remultiplication. The remultiplication of planets in a star system happened by them coalescing simultaneously out of a gas cloud. The remultiplication of cells in a body happened when one cell started dividing and the daughter cells stuck together. The remultiplication of bricks in a wall happened because a bricklayer iteratively put them there. The remultiplication of distributaries in a river delta happened because of iterated sediment deposition and channel switching. Regardless of the mechanism, there is a recurring compositional theme in all of these, and I call that remultiplication.

Remultiplication and fractals

“Fractal” is another term used for entities with internal similarity. Whereas remultiplication is concerned with similarity on one level of a partonomy, fractals also exhibit similarity across different scales and thus up and down the levels in a partonomy. It is possible for there to be up- and down-scale self-similarity without remultiplication, such as in a logarithmic spiral, whose zoomed-in center is self-similar to the zoomed-out whole.

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Definition

A metatope is a mathematical object constructed by a layering of spaces. For any space in a metatope, points can be plotted in it. These points in the space correspond to either (1) datapoints from a dataset with a commensurate dimensionality as that space or (2) spaces that contain either (1) or (2) (potentially ad infinitum).

A directed graph can be constructed from any metatope. The spaces within the metatope map to nodes, and the connections from a space to the spaces plotted within it map to the edges. A “highest space” can generally be identified, and that is the root of the metatope’s graph. The degree of a metatope is the maximum distance in the directed graph starting from the root. The degree counts the number of layerings in the metatope: a metatope of degree- is just a normal space, and its graph is just a solitary node with no edges. If there is a root space in the metatope and all the other spaces connect upwards to exactly 1 space, then the graph of the metatope is a rooted tree.

Example

Consider a dataset of faces, both normal and cyclopean, parameterized by a robust set of features such as interocular distance, nose length, cheekbone position, etc. All the normal faces can be plotted in one space, and all the cyclopean faces can be plotted in one space, but these two spaces are incommensurate because there are several dimensions that are not shared (most saliently from the list above: interocular distance). Both spaces, however, can enter as points in a higher-level space that would have one dimension called “number of eyes.” This would be a degree- metatope, and its graph would have two nodes connecting upwards to the root node, for a total of three nodes.

Note that the structure of any metatope depends entirely on the relevant set of datasets (actual or theoretical) and the reckonings made in how to combine them, and that multiple metatopes can often be combined into one metatope, and that the same dataset can fit into topologically distinct metatopes that carve their dimensions, hierarchy, or even number of spaces differently.

Motivation

My motivation is to develop a mathematical construct similar to the mere space that can be applied rigorously even when dimensionalities are incommensurate. I have often encountered conversations where “the space of x’s” is tossed around even when the x’s obviously or probably differ in the number of dimensions from one x to another x, such as “the space of conscious experiences.” I hope that the construct operates as an intuition pump and leads to interesting insights, from myself or others.