Introductory Summary of Ideas in CTMU

There will be many perspectives that we will progressively identify through dualities. Each paragraph is an incomplete description; these will be glued together as one progresses. Mathematical analogies are unavoidable.

View space as a continuum.

Every subset of space corresponds to a cognitive-perceptual processor (also called transducer or self-transducing unit of information). Via a sort of philosophical limit of model theory, these "self-transducing units of information" are to be thought of as a self-modelling language -- a language which is its own model and interpretation thereof. It is illuminating to think of this as a sort of limit of model theory in which the model contains the first-order language and its interpretation therein, and the mechanism thereof, ad infinitum, in a comprehensive self-contained loop. Of course, in the process of this, the language becomes much more than first order -- it becomes its own model. Langan describes this as a sort of "directed limit of model theory", the directed limit being taken over higher-order models of successively greater self-simulative capacity. Any mathematical theory requires us to model it in our head, so is not descriptively self-contained, but at that ultimate limit of a perfect self-modelling language, there is a singularity in which the system internally models itself -- i.e. is conscious.

These units of information correspond to conscious beings -- and we will see that in CTMU, every subset of space is ascribed its own consciousness, and these self-transducing units of information (i.e. these self-modelling languages) distribute downward and glue upward, like a sheaf from algebraic geometry.

We shall hereafter refer to them as either "transducers" or "units of information". Do not think of a "unit of information" as like a 0 or 1 but as a self-modelling language. The "unit" part comes from the externalization, by which they can be (on one side of a duality) fed into other transducers as the system distributively self-refines.

Each unit of information and their interactions with each other can be viewed in various incompatible ways which we claim are ultimately isomorphic and identify; through this, various positions can be simultaneously true and false depending on in which side of the duality we phrase the notion. For example, self-determinism and acausality ("informational incompressibility", which scientifically corresponds to "randomness") can be dually identified as the internal and external perspectives. There are also trialities, and various dualities/trialities can interact to form higher-level polyalities. We can view these dualities as corresponding to automorphisms of a single higher theory. In fact, we can view them as corresponding to identified automorphisms of a unit of information, under which we identify generalized fibers (category-theoretic subobjects). Or in the language of model theory, we can think of a unit of information as homogenous and strongly saturated, whence identification of fibers under automorphism is equivalent to identification of types. It is important to note that this implicitly takes the philosophical stance of identification of indiscernibles, which is necessitated both by the foregoing and that in a unit of information the language is the model.

On one side, each transducer can be viewed as a static subset of space deterministically acted upon by its external environment and its own state, like a classical automata-theoretic cybernetic system.

On the other hand, we can view each transducer as completely self-contained, a full reality within itself. We view this self-modelling language as a dual-aspect monad containing, on the one hand, the "syntax" or rules of self-transduction and "state" or configuration. These two are how the system contains itself in a two-part way via topological-descriptive duality. The system topologically, or spatially, contains its description, which in turn descriptively contains the system. However, these two are "out of phase", with loose descriptors that require antecedents. This is the sort of stress on the system that causes it to progress forward in what we call time. The system fills loose predicates with arguments in the "descriptive containment" phase, resulting in the evolution of the model from one state to another, and this, in turn, is realized as the language itself in the "topological containment" phase (the model containing the language). It should be noted that both of these happen at once, in some sense in a single timeless moment (now), but are teased apart as dual aspects of a self-contained self-modeling language.

Because each state of the system is in fact also its syntax, it is more accurate to refer to the evolution of the system as an evolution of its syntax-state relationship. Topological-descriptive duality describes the alternating (yet simultaneous) mechanism that causes a system to timelessly self-refine, but as syntax = state this is really the syntax-state evolving as a single unit of information. Langan invents a term called "telesis" (adjective: "telic") for the mechanism by which syntax and state mutually refine each other. We can view telesis as what emerges from TD self-containment when we re-identify syntax and state, or equivalently identify the two containment phases. When we identify the two containment phases of reality, we get that it self-refines through telesis. Telesis has a sophisticated theory associated with it which we will far from fully explore in this post.

There is, in addition to the above two, an intermediate perspective. This view combines automata-theoretic transduction with telic evolution. We return to our view of units of information as being like a sheaf over continuous space endowed with its usual topology (less formally, corresponding to subsets of space). In the course of transduction, each transducer has a certain freedom over its next state. We view the external transducers as not entirely determining its next state, but rather restricting it to a certain set of outcomes from which it freely chooses its next state through telesis. This new state is then externally reflected as its new configuration and fed into the restricting mechanism on other transducers. Note that by this view, smaller transducers are more restricted while the universal transducer (the one corresponding to the totality of reality) is completely unbound -- hence why we refer to its telesis as "unbound telesis."

These various perspectives can be identified as all happening to perfectly coincide, whence they are dual. In one perspective, there is determinism, in other, the system is perfectly self-deterministic as it freely self-configures through telesis (or equivalently the alternative phases of TD-duality by which the language configures itself (and configures its configuration of itself, and of that, at the limit of which is what we can call "free will" or more accurately self-determinism)), and in the third, a hybrid synthesis of the two.

The currency of telic feedback is "generalized utility"; it is this which a unit of information seeks to maximize in the course of its telic evolution. Generalized utility is "autologous", or intrinsically defined within the system. In fact, the system and generalized utility mutually define each other, so generalized utility is autologous on the same level that a self-modeling language is autologous. But if generalized utility is associated with each potential future state of the system, yet is intrinsically defined within the system as it currently is, how can the system be perfectly self-contained if it can quantify over its own potential futures? The answer is that a unit of information contains its own possible futures through a principle called "hology" (a broad abstraction of the holographic principle) -- and in fact, this is of what it is composed.

Hology emerges at the most basic level from the fact that the predicates "reality" and "unreality" are mutually dual and mutually define each other as mutual negations. They are two sides of the same coin. If we imagine a transducer (in the "hybrid perspective") as the region inside a sphere corresponding to its boundary,  the boundary is the boundary between reality and unreality. (Each unit of information is a self-contained reality.) It is this boundary that matters; what is external and what is internal are informationally identical (both contain all the information of reality -- be it as itself or as what it is not). Thus, we identify what is external and internal to the boundary, and can think of what is external to a transducer as holographically projected within the transducer as its own contents. We can imagine this like identifying points in the complex plane with their images under the inversion map that sends x to 1/x, where the unit circle is the boundary.

By hology together with the hybrid model, a transducer is distributively composed of its own possible futures as its own contents. However, its own possible futures are equi-informational with the configuration of the external system of transducers -- i.e. of the external universe. Because these two are informationally equivalent (again, swapping the two can ultimately be done through an automorphism of the global structure), it is in this sense that the contents and structure of a transducer are of that external to it. Thus its distributed contents are its environment, and are also its future together with others that it sheds. Within a unit of information, these are like the sentences of its language, which are themselves units of information which can be dually regarded both as substructures of and as external elements of itself.

Through hology, each system is composed of that external to it -- in some sense, the rest of the universe, But this means that telesis maps a system within itself -- an "endormorphism." On the other hand, by that same hology principle, it is telically mapped to some state external to itself -- an "ectomorphism." On the endorphism side, regions of space and objects are continually being mapped within themselves, resulting in the evacuation of space and their "getting smaller", which amounts to the expansion of space relative to them. (Thus we see that the internal contraction of transducers is dual to the expansion of space). On the ectomorphism side, objects move through or change in space. And both of these are but two dual aspects corresponding ultimately back again to the necessary mechanism of the comprehensive self-containment of some (any) self-modelling language. (This follows more generally from the metalogical "multiplex unity" principle -- which we will describe -- together with the a priori necessity of hology.)

When we identify this continual ectomorphism and endomorphism as mutually self-dual, we call the unified principle "conspansion" (Langan's etymology: portmanteau of "material contraction qua spatial expansion"). One can see from the above that conspansion is ultimately equivalent to distributed telic evolution.

Now, conspansion happens at a single rate -- the rate of exchange of information within/without, which is identical to the rate of the alternating TD self-containment. This rate is necessarily intrinsically defined and constant (and it is always ever relative to itself), and corresponds to the speed of light, i.e. the speed of causation, and being intrinsically defined is necessarily invariant across all reference frames.

Furthermore, this constancy together with the endomorphism phase of hology have an interesting consequence when we consider relative scales of units of information (e.g. a transducer within a much larger transducer). Conspansion is constant, but being intrinsic is relative to size (small and large are obviously relative to each other, but large transducers will conspanse at the same rate *relative to themselves* as small ones do relative to themselves). Thus the scale of expansion is proportional to size, and thus exponential as a function of space. Langan says this is the a priori justification for the positivity of the cosmological constant.

When it comes to the telic evolution of systems, it is important to remember that by the sheaf-like nature it is both downward-determined and upward-determined as the "gluing" of the evolving units of information of its subsections. The hological restrictions that the external=subcomponent configuration of the universe place on the state=future configurations of a transducer are precisely those that ensure transducers' distributed coherence as a function of time, i.e. preserves the "gluing axiom." Self-determinism is free precisely up to preservation of this schema.

Remember, view all of these dualities as simultaneous. After all, we are teasing apart a singularity, the dual-aspect monad of a self-contained self-processing language -- of some(any) reality (i.e. conscious experience thereof). There are so many corollaries, including that self-processing units of information are distributively composed of self-processing units of information. This is in fact Langan's "multiplex unity" principle, which Langan assumes as a priori (but it is no surprise that other aspects of the theory re-imply it). We can see from hology, telesis, and multiplex unity how a unit of information, in Langan's words, "globally self-transduces without input."

This is all only the beginning, I have introduced only a fraction of the contents. We have barely even talked about telesis, though it is at the nucleus of the CTMU. Though self-determinism is a necessary property of reality by virtue of reality having nothing to determine it but itself, we have scarcely seen the (ultimately cyclical, self-defining) nature of telesis by which it operates. And though we have seen some of the dualities imply aspects of general relativity and even the positivity of the cosmological constant, we have not seen the role of or a priori necessity of aspects of quantum mechanics. We have also not seen many other vital duality principles. Perhaps most of all, we have not seen the nature of or necessary properties of the universal self-transducer -- what Langan personally identifies with God.

But I hope this is a starting point.