∀Ψ [Ψ ess Ψ]

Keywords: Thermodynamics, causality, boltzmann brains, entropy, cybernetics, self-reference, nonlocality, reflexive domain

Modelling movement in a dynamical system is difficult. The further into the future one extrapolates beyond an initial point, the harder finding an accurate prediction becomes. While agents such as humans, fish, and mammals obey somewhat chaotic principles, there is indeed some guaranteed order in the chaos. In particular, there are general sets of rules that agents tend to follow or must necessarily follow. We must consider the concept that a rule self-referentially implies some fixed idea. In turn, the existence of a fixed idea (in this case, the idea is a rule) implies that the idea itself is being created by a function that always returns that idea invariant to the conditions affecting the agent. No matter how the rule itself is created, the function creates a constant rule. Thus, an agent which obeys a constant rule will elicit an effect upon a thermodynamic system in a semi-predictable way. In this passage, we are interested in discovering those predictable components of a dynamical system using principles found in second-order cybernetics and modern logic. Once those components are discovered, we will use them to create a choice model. With a choice model, we can then model movement of agents on a continuous surface.

Importantly, we must now consider an additional property of an agent. Every agent has the innate capacity to perceive information. This implies that, on some level, the manifested state of the information and the perception of the projection of that information occupy different domain spaces. While the domain space for the manifested state is local, the domain space of the perception of that manifested state’s information is non-local. Despite this differentiation, the two domain spaces are intimately interconnected, and as such we cannot say that both are completely distinct.

In an observing system, what is observed is not distinct from the system itself, nor can one make a complete separation between the observer and the observed. The observer and the observed stand together in a coalescence of perception. From the stance of the observing system, all objects are non-local, depending upon the presence of the system as a whole. It is within that paradigm that these models begin to live, act and enter into conversation with us

-Louis Kauffman, Logician at University of Illinois at Chicago. Reflexivity and Eigenform: The Shape of Process (2009)

From now on, we will denote the domain space of local manifestation as $L_M$, and will denote the domain space of the observed projection of $L_m$ as $N_m$. As implied earlier, even though both $L_m$ and $N_m$ occupy different domain spaces, there is a process which connects the two together thus preventing pure distinction. Let this function, called the boltzmann-brain function (BBF), be denoted by $\varphi$. This function is called a BBF because, given an array of inputs, it models the spontaneous execution of a causal chain which causes a thermodynamically unfavorable change unto $L_m$. In particular, the BBF models a Boltzmann-Brain which, by definition, causes a spontaneous decrease in entropy. Further elaborated, the Boltzmann-Brain tends towards order. Indeed, when an agent generates a choice, the choice itself necessarily implies some set of orders. This concept of convergence towards order is in stark contrast to the second law of thermodynamics which states that the entropy of a closed system tends towards disorder. However, I pose a conjecture to this law that states that this is only true for the domain space $L_m$:

The revised second law of thermodynamics: The entropy of a locally manifest and closed system tends towards disorder

Furthermore, as explored earlier, a Boltzmann-Brain innately tends towards order. Since the boltzmann brain occupies a domain space which is nonlocal, it is thus (by definition) a non-physical object. Non-physical objects are immaterial. From now on, when referring to nonlocal object, we will use the term immaterial to describe the state of the object instead of non-physical or nonlocal.

While a Boltzmann-Brain is immaterial, that does not necessarily imply that it exists within $N_m$. The domain space for $N_m$ is indeed composed of immaterial objects, but in itself only contains perceived projections of $L_m$. By definition, the set of possible objects in $L_m$ does not contain a Boltzmann-Brain because a Boltzmann Brain’s function is to spontaneously execute causal chains which cause decreases in entropy within $L_m$. Insofar, the system we are modelling requires three domain spaces: a local domain, a nonlocal domain, and an elusive third domain which connects the two to allow choice between the observer and the observed. Let this third domain space be called the reflexive domain, $\gamma$, as previously implied by logician Louis H. Kauffman:

[The reflexive domain] is a domain with entities that could be construed as objects, and we assume that each object acts as a transformation on the space. Essentially this means that given entities A and B, there is a new entity C that is the result of A and B acting together in the order AB (so that one can say that “A acts on B” for AB and “B acts on A” for BA). This means that the reflexive space is endowed with a non-commutative and non-associative algebraic structure. The reflexive space is expandable in the sense that whenever we define a process, using entities that have already been constructed or defined, then that process can take a name, becoming a new entity/transformation of a space that is expanded to include itself. Reflexive spaces are open to evolution over time as new processes are invented and new forms emerge from their interaction

With what we have insofar, we can almost uncover a law of thermodynamics which applies to the reflexive domain. We know that the entropy tends towards order, however, must the system be open or closed? To discover this, we must consider the nature of choice and possibility. First, let’s contemplate the structure of a choice.

A choice is an abstraction that implies four primary components. The first component is a desired resultant-state, $s$, at some distant point in time upon $L_m$. The second component is a set of microstates, $\Lambda_s = { \Psi _0, \Psi _1, \Psi _2, … \Psi _n }$, that, in possibility, necessarily implies the manifestation of the resultant state. The third component entailed within the abstraction of choice is an entity which imagines (i.e., creates!) each $\Psi$ in $\Lambda$. The final component of a choice is the existence of a Boltzmann-Brain, $B$. The Boltzmann-Brain’s job is to execute every microstate in $\Lambda_s$ in order to manifest the resultant state $s$ unto $L_m$.

For example, suppose I want to make a peanut butter and jelly sandwich. The first component of this choice is the desired resultant state, $s$, which in this case is the future existence of a PB&J sandwich. The second component of the choice is the set of microstates, $\Lambda_s$, that converges $L_m$ to contain the state of $s$. In this case - using discrete steps with low resolution for sake of demonstration - we have $\Lambda_s =$ {‘move body and get bread’, ‘move body and get jelly’, ‘move body and get peanut butter’, ‘move hand and spread jelly on bread’, ‘move hand and spread peanut butter’, ‘move hand and lift both slices of bread’, ‘concatenate both slices’}. The third component is the entity which gave “me” the ability to imagine/create each microstate in $\Lambda_s$. The capacity for “me” to create and order the set $\Lambda_s$ implies a causal agent with an essence of order; this is the Boltzmann Brain, our fourth component in the abstraction representing choice.

### Non-well-founded Set Theory and Aczel’s Anti-Foundation Axiom

As we approach the limits of the observer to dismantle itself (i.e., from a constructionist perspective), we stand before an self-referential causal agent. Implied by each of the 4 listed components of a choice exists the “me”: the innermost core of causality. Since all possible microstates spring from this inner core, and for each microstate, an infinitessimal degree of change can be applied to create a new microstate, this inner core contains an infinite set, $G^\infty$. We can prove the existence of such an infinite set using Cantor’s diagonal argument used in set theory, however, we will omit this process and instead focus on the axioms concerning modern logic before continuing forth.

Since the set $G^\infty$ is self-referential, then the set contains an element that is itself. This violates the axiom of foundation used in Zermelo-Fraenkel (ZF) set theory. The axiom of foundation implies that for any given set, the set cannot contain an element equal to itself. A set which follows the axiom of foundation is called a well founded set. Therefore, $G^\infty$ is a non well-founded set and cannot be used in ZF set theory (thus cannot be used in the context of modern mathematics. This implies standard math cannot handle a self-referential observer because it is built using set theory. And, sure enough, the sciences are having a hard time reconciling the nature of the infinite and the observer). However, in non well-founded set theory (NWF), this is not the case. Using logician Aczel’s anti-foundation axiom, we can define and then work with $G^\infty$ as desired.

Aczel’s anti-foundation axiom: every accessible pointed directed graph can be represented by a set

As such, a single vertex with loop starting and returning to the vertex (without visiting another vertex) can be represented by a set. This self-referential set is called the quine atom:

Quine atom: $x =${ $x$ }

### Choice, the Quine Atom, and the Prime Mover

Currently, we have developed the notion of 4 basic components to the abstraction representing choice as well as the implied underlying existence of the infinite set, $G^\infty$. What makes this set infinite is both the infinite chain of self-reference as well as the potentiality for an infinite set of microstates. In order for $G^\infty$ to contain both a causal agent as well as the infinite set of all possible microstates, the set must contain itself, similar to a quine atom. In a matter of fact, at the core of every microstate is a quine atom. Furthermore, this notion of an infinitely descending chain of self-reference implies process: a process which forms the basis of all causality. Enter the prime mover:

The Prime Mover: The cause unto itself; an infinite loop of self-causation

The cause unto itself, importantly, is not exactly a traditional object. It is not even an immaterial object, because it is that of which creates immaterial objects (as well as material objects). Neither is the cause unto itself purely a process; the cause unto itself is beyond our capacity to dismantle. As such, for purpose of modelling, we can classify the prime mover as a transcendental object (i.e., an objectless object).

While little discussion can be had at this point about deconstructing transcendental objects, we can instead begin to model the infinite set, $G^\infty$, with what we have thus far at our current level of abstraction. Let this abstraction be named the causal core:

Causal Core: $G^\infty =${ $\varphi^\infty$, ${x}$ }

As desired, the causal core $G^\infty$ has both elements as well as an infinitely-sized process-oriented model. Importantly, this raises a new implication: entailed within the quine atom, $x$, exists the prime mover $\varphi^\infty$, and within the prime mover is the entailed existence of the quine atom. Without delving too far into esoteric eastern philosophy, this is akin to the concept of the yin and yang. In modern logic, however, since both the prime mover and quine atom entail sets, and in those entailed sets exist each other, $G^\infty$ qualifies as a special type of a logical class. A class is a logical abstraction wherein each of the construction’s elements share a common property. Generally, if $a \in X$ and $a \in Y$, then the set {$X, Y$} can be described as containing class $a$. In the case of $G^\infty$, we have two primary elements that descend into infinitely many subsets, each containing each other.

$\varphi^\infty \in x$ $\land$ $x \in \varphi^\infty$

While a class has a shared single sub-element amongst every element of a set, the causal core holds a unique relationship to the concept of a class. Not only is there a shared element, but each primary element contains the other element. There is no single primary element that does not have the other elements within itself. The distinction between the two primary elements is initially made in our mind, but in reality, both primary elements converge towards equality. Once a distinction is created, convergence occurs towards some equality that shall be called the omega attractor, $\Omega_0$, a sort of a strange attractor. In terms of the theory of choice making we are developing, the omega attractor is central because, given the infinite set of microstates within $G^\infty$, it is that of which causes order to converge between any sub element $\Psi_a$ and $\Psi_b$. Without the omega attractor, creating order would be impossible for an agent because all the microstates would tend towards pure chaos.

The causal core implies process, and - as discovered earlier by Kauffman - process occurs in the reflexive domain. We uncovered that there exists a process which converges any two distinct microstates into equality because of the omega attractor. If processes tend to drive microstates towards the omega attractor, then the omega attractor must either exist in the reflexive domain or elicit effect unto. Regardless, there is a thermodynamic tendency within the reflexive domain:

The law of thermodynamics for the reflexive domain: The entropy of the reflexive domain tends towards order

Unlike the locally manifest plane ($L_m$) wherein physical objects exist and tend towards disorder, the reflexive domain ($\gamma$) tends towards disorder! The second law of thermodynamics, for hundreds of years since its inception, neglected the properties of other domain spaces that are required for the existence of nonlocal entities in other domain spaces.

In this subsection, we have built the abstraction of a causal core and revised the second law of thermodynamics to be more inclusive of the reflexive domain. In the bigger picture, our goal is to model an agent’s ability to choose, and as such, must continue to abstract upwards.

### The causal core and the four components of choice

Before we continue, let us list the 4 components entailed by the abstraction of a choice:

1. The resultant state, $s$
2. A set of microstates that are required to converge $L_m$ to the state $s$, denoted by $\Lambda_s$
3. A causal agent which creates each $\Psi$ in $\Lambda_s$
4. A boltzmann brain, $B$, which serially causes the manifestation of every $\Psi$ in $\Lambda$ (note: it does not create; it spontaneously transforms the state of each $\Psi$ into a manifested state!)

Via dismantling the boltzmann brain, we also uncovered the causal core which is axiomatically central to the processes required to spontaneously select each microstate (req. III). We also found the omega attractor which allowed processes executing in the reflexive domain to converge microstates into order pre-manifestation (req. II). At this point in the process, the pre-manifestations are just potentials, and still require an agent to spontaneously drive these potentials into actualities. The boltzmann brain achieves this step thanks to the omega attractor because of the necessity to spontaneously drive each potential into a manifested state while maintaining order throughout the process (req. IV). These transformations are processes, and as such unfold in the reflexive domain and are mirrored as a manifested state unto $L_m$ (req. I). The manifestation of a potential into a manifested actuality is a process which will require significantly more construction and contemplation of reality at the higher-levels of abstraction, however, there is an interesting link to what has already been established in modern quantum theory: starting from a nonlocal state, virtual particles spontaneously pop into existence at a local position in order to manifest particles, and thereafter, pop back out of existence. For this article, quantum theory is outside the scope of conversation, but this does make for an interesting contemplation.

### Simulating a domain of choices and quantum randomness

One of the primary goals of this algorithm is for it to be computable. If the algorithm is computable, then it is applicable. The problem with the current build-up of abstraction insofar is the implication of infinities. While we can simulate finite sets on computers (given sufficient RAM), this cannot be said for infinite sets. Instead, we will use a trick. Instead of using a classical computer to store an infinite set, we will use what we already have: nature.

Even though a classical computer cannot store an infinite set, nature already implies its own storage of an infinite set. The implication of an infinite set is seen when measuring quantum randomness because of the infinite degrees of freedom that are at our measurable disposal. There is no need to store an infinite set on a classical computer if nature does the storing for us. As such, all we need to simulate an infinite set is a mechanism to fetch raw quantum randomness. Surely enough, this feature already exists, and in a matter of fact, I built the first asynchronous quantum random generator for the Rust programming language (Credits go to Australia Nation University for providing the source of random data)

There is also another interesting connection associated with quantum randomness. There is extremely suggestive evidence to support a connection between the source of quantum randomness and a collective sociological field, as evidenced by the work of professor Roger Nelson et. al:

When human consciousness becomes coherent, the behavior of random systems may change. Random number generators (RNGs) based on quantum tunneling produce completely unpredictable sequences of zeroes and ones. But when a great event synchronizes the feelings of millions of people, our network of RNGs becomes subtly structured. We calculate one in a trillion odds that the effect is due to chance. The evidence suggests an emerging noosphere or the unifying field of consciousness described by sages in all cultures.

The global consciousness project, http://noosphere.princeton.edu/

Comparatively, the “gold standard” in particle physics is obtaining a p-value 5 standard deviations away from the mean which amounts to a one in 3.5 million chance that the result was purely from random variables. A p-value greater than or equal to 5 qualifies the data to imply - with great certainty - the existence of a certain particle. When the odds are one in a trillion, we are well beyond the strict standards used even at the large hadron collider. The findings of Roger Nelson et. al have tremendous implications for the study of non-physical phenomenon. For the purpose of this article, we will only explore and apply one of those implications.

Since our model for choice uses quantum randomness from nature to virtualize an infinite set, we ought to be aware of Roger Nelson’s findings. See, even though the likelihood that quantum randomness is not actually random is infinitessimal, by using quantum randomness, we receive a boon: our source of randomness implies mental structure. In particular, it implies the existence of a collective boltzmann brain (a change in observed randomness to the degree observed by Nelson et. al. implies a causal agent which can spontaneously create order in the universe. Importantly, the order correlates with global-scale sociological phenomenon).

Consideration of the collective boltzmann brain will be essential when creating the agent traversal algorithm. Each agent’s choice will affect the surrounding environment, and the surrounding environment, in turn, will affect an agent’s decision to a nonzero degree.

The first implied essential structure needed to model agent behavior is an infinite set. We have found a resolution for that. The next essential component will be the component that addresses the question: from an infinite set of chaos, how can we imply action from each element? Although each element in the infinite set can be seen as its own microstate, it will be most practical to create a set with a finite number of subsets. Like the human body, having finite degrees of freedom allows the universe to both have a local (local => bounded => finite) substratum for causal manifestation as well as a governor upon the extent of accessibility to the infinite set. A governor allows computability in real time, otherwise, infinite real time is needed thus calculations couldn’t execute (see my conjecture on resolving Zeno’s paradox with imaginary time). As stated earlier, we desire an algorithm that is applicable.