The lights are on, or the lights are off. On-off is a binary description standing above more technical 'underlying' descriptions: We know that really there is a light and a switch; the switch's state has a range of angles, corresponding to the light's on-off state, unless it is broken, etc. Further still: The light and switch are material things, with material states; continua of matter, with internal and external forces. The angle of the switch is materially and causally linked to a copper wire, a medium for a gradient of electric potential which leads spatially to the light, affecting its state in a particular way, depending on its current state (eg if it is broken the switch does nothing). Further still: Material things are not continua, but swarms of particles with positions and momenta, or particle-like patterns of a wavefunction, or particle-like patterns of field-like patterns of a wavefunction, and so on. As the resolution of the description increases, we dig down closer to some raw reality. Or we might turn around the other way, toward abstraction, to compress information, but along a different path, say, seeing the switch and light as elementary units in a circuit diagram... Or we might broaden the domain of our description, placing the light within some larger structure, as a purposed object used to lighten a room, or as an economic good with an exchange value, or as a particular make and model of technology, or other various symbolisms...

Thus unfolds a space of representations, fully traversable provided one has the appropriate knowledge of each description, and how it relates to each other. The on-off binary, despite its lackluster single bit of information, dominates practical life. And justifiably so- when the light behaves 'as intended', as is typical, the on-off binary is sufficient. We need not know the internal details, perhaps we need not even situate it in time and space... until something goes wrong, "[raw] reality bites", and we are forced outside the domain of this abstraction, forced to open a blackbox or two, to work with whatever underlying abstractions implement the binary, or (we hope not) whatever abstractions implement those abstractions, and so on.

There is an apparent directionality, a 'degree of abstraction' one might assign to different descriptions. There is sometimes a (mistaken) temptation, even, to insist on a linear chain of more or less abstract models. Does this chain have ends? Is our 1-bit on-off description the most abstract possible? What about the least abstract? Is it abstraction all the way down, or is there something at the bottom? What are the limits of this analysis?

I reject the linear-chain intuition, but I nonetheless offer two opposite 'limiting cases' of abstraction.

1. The 'most-abstract' representation I will call the terminal abstraction¹. It represents no information, no particularity, serving only to indicate². It says only "there is," capturing nothing whatsoever about what is, or how it is is-ing. When we give things names, in an ideal sense, without saying anything about them, we are applying onto them the terminal abstraction, to simply say "here it is". (Though from a less abstract view, the particular names we assign will inevitably situate their objects within some larger system of meaning...)


2. The 'initial object' at the 'bottom' of abstraction I will call the initial abstraction, or more evocatively, raw reality. This is the noumena, the hypostasis, the nonrepresentational, the Real, the "Reality By Itself" of reality realism, the ultimate substance at the bottom of form, the ultimate territory underlying all maps, and the singular concretion beneath all abstraction. While the terminal abstraction is an all-encompassing blackbox, the initial abstraction is the lack of any box at all, of any identification or system of coordinates or conceptual framing to get ahold and situate oneself and one's experience. We do not have 'access' to raw reality: We can say nothing about it except through its relations to our conceptual or sensory systems, our abstractions.

These two objects are not two ends of a chain, but rather, points of convergence and divergence in the infinite omnidirectional unfolding of abstraction. There is a space of abstract objects, of stracta: Conceptual or mathematical systems, representations, generalizations, models, theories, maps, minds, frameworks, ontologies, structures, constructs, patterns, paradigms, narratives... Stracta are connected by embeddings: Relations of implementation, homomorphism, approximation (approximorphism), application, representation, compression, or simple surjection, projection, recording, encoding... An ab-stracta is embedded in a sub-stracta³: A jpeg approximates a png, fluid mechanics emerges from particle motion, the integers are a (additively closed) subset of the reals, information is communicated with linguistic structures, group theory is represented with linear algebra, frays are pretended by thrones, python is written in c which is written in assembly, and an on-off binary captures something of the typical behavior of a material light and switch.

These stracta are not raw reality, but neither are they necessarily unreal- real patterns genuinely capture features of that which they abstract. Raw reality is real, and it really implements patterns, regularities⁴ from which unfolds the larger space of abstraction. Thus emerges a hypergeneralized representation theory, a metaphysics of patterns, which seeks to accommodate those sciences which rebuild their ontologies from scratch with each new development.

When two stracta genuinely capture all the same features, such that they can be losslessly substituted, they are isomorphic. But more usually, there is always some loss, some leakage or overflow of representation, something lost in translation. Here is my attempted taxonomy of overflows; a taxonomy which is itself overflowing quite a bit, as the categories bleed together at the borders.

• Domain overflow: Each embedding has a domain, the 'part' of a substracta to which an abstracta applies- economic theory says nothing about black hole formation, for example. When wandering outside of a stracta's domain, one leaves the regime where the stracta is implemented. When approaching the speed of light, one wanders outside the domain of Newtonian mechanics, and must embrace the deeper relativistic framework. When \(x\) is large, it can no longer approximate \(\sin(x)\). Domains may have rigid bounds, or gradual tails, with error building the further one wanders off.


• Entropic overflow: Abstractions are often coarse-grained, leaving some range of 'in-between' which escapes rigid categorization- Everywhere there are heap paradoxes. When one says "red", another might ask, "which red?" because there is a broad range of underlying colors which are being conflated by this system of representation. And how far does this range extend? Does it include maroon? The total entropy of an embedding is a measure of its entropic overflow, and the entropy an element of an abstracta is a measure of the entropic overflow relevant to that element. In surjections, where the labels have sharp definition, Boltzmann entropy will do; otherwise, the 'fuzziness' can be handled probabilistically by Shannon.


• Estimation overflow: Since embeddings need not be strictly homomorphic, they may have a simple approximation error. For homomorphisms, which have no estimation overflow, there is a "law of composition", where some tip-to-tail embeddings \(e\) and \(m\) imply there exists some other embedding \(e\circ m\), which is simply their combination. But approximorphism composition has compound loss: \(a\approx b\approx\ldots\approx z\) does not imply \(a\approx z\).

From the concept of embedding follows four ways for a stracta to be related to another: As isomorphism (symmetric equivalence), as abstracta (the relative 'abstraction'), as substracta (the relative 'concretion'), or as unrelated (no embedding). Thus there are three kinds of moves one can make when traversing the space of abstraction:

1. Reframings are 'neutral' changes of perspective, tracing lines of isomorphism (eg the move from Newtonian to Lagrangian to Hamiltonian mechanics).


2. Constructive movements work 'upward' toward abstraction. Perception without heavy abstraction is overwhelmed by raw data (autistic overload), thus minds function by congealing patterns of sensations into predictive stracta. The conscious mind can then work with simpler high-level abstracta, both perceptually and behaviorally, which are implemented by lower-level unconscious substracta. I think something like "I'm going to go over there", which my unconscious maps onto lower-level operations which handle all of the complexity of moving my legs and so on. Overflow usually has a negative connotation because it indicates the breakdown of a useful tool (hence the negativity in my description of blackbox-opening in paragraph 2). Lightweight higher-level abstractions are useful, necessary even, for practical modeling, and thus for effective centralized governing (law of requisite variety)... If only the relevant systems would hold still in the domain of such abstractions.


3. Deconstructive movements work 'downward' toward raw reality. This is, perhaps, the Kuhnian scientific process, the dialectical synthesis, and/or Nick Land's invasion of the outside. In physics, overflow is obsessively sought out and resolved, subsumed by some deeper theory with a broader and/or more accurate account. Competing paradigms are found to be (if not simply incorrect) different domain-specific approximations of some underlying paradigm, which implements the old paradigms via some "correspondence principle". In general, deconstructive movements are breakdowns of macro-patterns into their constituent micro-patterns, the dissolving or deterritorialization of coherent structure into a more complex field of 'constituent' operations.

Given a stracta, one can ask of its capacity as an embedding medium and the flexibility of its implementation; that's to say, of the range of stracta embedded in it, and the quality and complexity of those embeddings. "The medium is the message," insofar as the substracta determines the abstracta, and different embedding media have different embeddings with different overflows. Set theory works so well as a foundation because it losslessly implements all of mathematics from simple axioms. Apparently language and human minds are similarly powerful, seeing as they can encode/convey all of this, and seeing as they implement set theory, though not without overflow (linguistic ambiguity, forgetfulness, etc), and only within particular domains of operation which do not necessarily come easily. The way such substracta (media) are embedded in our language and minds determines the way we operate with them, thus even isomorphic stracta may have different mental/social consequences, in the same way that different sorting algorithms have different implementations with different runtimes and memory usages despite their isomorphic functionality.⁵ Of interest is the capacity of social fields, with their embedded social constructs, and the relation of their capacities and overflows to phenomena of oppression and whatnot. Minorities generally appear as victims of abstraction (eg neurodivergence as overflow of folk psychology's "intentional stance").

~

Anyway, this is all a sort of development of structural realism I've been trying to think through for the past year or so. I have sometimes called it "abstraction realism", because I find that name funny (since "abstraction" is colloquially defined by its unreality), and because it captures a sort of aesthetic appeal the whole thing has for me. Most of what I read on structural realism is more interested in defending the view against other realisms than in actually developing some practical system, and I'm much more interested in the latter, since a generalized ability to trace relations between abstractions seems incredibly useful. I thought I might find the more practical interest studied in category theory, but that hasn't really worked out so far.

The system I've laid out here is certainly not as 'tight' or rigorous as would be ideal- I haven't really probed the limits of what can be considered a stracta or embedding, nor have I been particularly systematic in my taxonomy of overflows. I've intentionally taken a very broad scope, but there is a fear that it is too broad, or too vaguely defined to work with. But even so, there is clearly a common thread pulled on by a wide variety of disciplines- homomorphism, media encoding, linguistic representation, structuralism, physics emergence, software abstraction, social constructivism... -and that shared thread is what I want to capture, to find and develop a general underlying framework.

How to overcome these problems? I don't know to what extent the answers are 'out there' somewhere for me to find by reading more, or if I should prioritize thinking through things on my own... I suspect mostly the former, since although I've had difficulty finding writings which address what I'd like them to, the odds nonetheless seem slim that I, who knows so little about metaphysics, can seriously propose and develop a metaphysics that hasn't already been worked through better by someone else. But I can attach my own silly words to things, at least.

Anyway, I should go and read more books, I guess.

~

1. The "terminal" and "initial" terminology is taken from category theory, but only in a loose way. The category theoretic definitions do not apply.
2. Possibly related to Spencer-Brown's calculus.
3. Ab- and sub- stracta are defined only with respect to a particular embedding. I do not mean to imply that certain stracta are fundamentally 'embedders' or 'embeddeds'; all are both, with respect to certain embeddings, with the exception of the terminal and initial abstractions which are purely ab- and sub- respectively.
4. We can (I think) be Humean about stracta in general. When objects are patterned by some 'generative' laws, the law is a separate stracta which the patterned object implements (eg when a set is defined by some rule, we can say that the set is a 'static' thing which implements the rule as a pattern).
5. Is this a contradiction? Should I say that they are not fully isomorphic, that there is some different sort of overflow, then? I'm not sure.