All authors contributed equally to this work. All authors contributed equally
to this work. All authors contributed equally to this work.

1]University of Sydney, Sydney, Australia 2]RWTH Aachen University, Aachen,
Germany 3]Pompeu Fabra University, Barcelona, Spain 4]Wolfram Research, USA

# Ruliology: Linking Computation, Observers and Physical Law

Dean Rickles<EMAIL_ADDRESS>Hatem Elshatlawy<EMAIL_ADDRESS>Xerxes D. Arsiwalla<EMAIL_ADDRESS>[  [  [  [

###### Abstract

Stephen Wolfram has recently outlined an unorthodox, multi-computational
approach to fundamental theory, encompassing not only physics but also
mathematics in a structure he calls “The Ruliad,” understood to be the
entangled limit of all possible computations. In this framework, physical laws
arise from the sampling of the Ruliad by observers (including us). This
naturally leads to several conceptual issues, such as what kind of object is
the Ruliad? What is the nature of the observers carrying out the sampling, and
how do they relate to the Ruliad itself? What is the precise nature of the
sampling? This paper provides a philosophical examination of these questions,
and other related foundational issues, including the identification of a
limitation that must face any attempt to describe or model reality in such a
way that the modeller-observers are included.

###### keywords:

Rulial Space, Hypergraph Rewriting Systems, Pregeometric Physics, Observer
Theory, Second-Order Cybernetics

##  1 Introduction: Foundations of Physical Theory

> If we had invented the digital computer before inventing graph paper, we
> might have a very different theory of the universe today.
> Jacques Vallee, _Dimensions_ 111 _Dimensions: A Casebook of Alien Contact_
> (Anomalist Books, 2008, p. 287).

Often when thinking about the modeling of reality in physical theories, we
employ an abstract space that is supposed to represent all possible states of
a system, a modal arena, one point of which will correspond to the present
state of a ‘real-world’ system. This provides the kinematical structure of a
theory when forces are ignored (yielding a larger space of possibilities than
is physically allowed) and the dynamics when forces are included (yielding the
so-called nomologically posssible states). In general, what is not in the
space is not a possibility; and what is not a possibility is not in the space.
This can be a _universal_ state space as in the geometrodynamics of John
Wheeler, where it is known as “superspace” wheegrav . Here, ‘points’ of the
space are 3-dimensional geometric configurations (i.e. Riemannian geometries
on a 3-manifold) of the universe and histories are then represented as
trajectories (paths through the space, generating spatiotemporal worlds),
corresponding to possible universes—the space of 3-geometries is understood as
the space of 3-metrics ‘quotiented’ by the diffeomorphism group (the
invariance group of general relativity), identifying those metrics differing
by elements of that group.222This is formally akin to the notion of moduli
spaces in algebraic geometry, where points in the moduli space correspond to
isomorphism classes of algebro-geometric objects and trajectories yield a
formal notion of dynamics on the space. Moduli spaces are useful for
classification problems, where coordinatizing the space is useful for studying
various classes of deformations (of the moduli parameters) corresponding to
the objects in question (see mod  for a useful introduction). Furthermore, in
quantum geometrodynamics, we envisage a wave-function over this configuration
space which assigns amplitudes for the various types of state of the
universe.333This is the infamous Wheeler-DeWitt equation with its problematic
interpretation in terms of dynamical evolution resulting from the absence of a
time-parameter $t$, itself stemming from the fact that time evolution is a
kind of diffeomorphism in general relativity and so belonging in the category
of gauge degrees of freedom rather than the structure to be assigned a
physical interpretation (see rick1 ). While presented as a rather fundamental
description of physics, even the superspace point of view clearly stands
several rungs up on the ontological ladder, presupposing several layers of
deeper structure.444This holds also for the ‘upgrade’ of quantum
geometrodynamics from 3-geometries and their histories to spin-networks and
their spin-foams, though the latter formalism features a slightly more
primitive structure in that it involves abstract graphs as its fundamental
objects and whose relations build up the various layers of structure we
associate with out physical theories.

The “Physics Project” recently initiated by Stephen Wolfram nks ; w1 ; w2 ; w3
; swcon  aims to describe how all other levels of structure are built from the
ground up, that is, from ontological ground zero. The basic structure is not a
set of elements as such, but a _totality_ that can then be decomposed to
generate possible universes, including our own. This structure is called the
“Ruliad” or “Rulial space” and is usually expressed informally as the result
of carrying out a process (or, rather, many such) to infinity, yielding “the
entangled limit of all possible computations”: it is what is generated by
carrying out all possible rules in every possible way ruliad ; swtm . It is
computationally exhaustive. Like the universal state space of geometrodynamics
or moduli spaces, sections of the Ruliad correspond to possible histories of
universe (though unlike the former two, the Ruliad is a purely syntactic
structure, defined independently of any _a priori_ geometric notions).

_Ruliology_ , a term coined by Stephen Wolfram, delves into the intricacies of
rule space and how different rules can lead to diverse and complex behaviors.
It represents the study of the Ruliad, a profound and encompassing framework
within the Physics Project that serves as the theoretical foundation for
understanding the myriad of computational universes. Instead of treating
reality as a mere collection of isolated entities, Ruliology embraces the idea
of the Ruliad — a vast, interconnected web of all conceivable computations,
executed through every possible rule. This intricate and boundless space is
not just a speculative novelty; it provides a comprehensive map from which
individual universes, including ours, can be derived. Such a perspective
challenges traditional views on the nature of reality and paves the way for a
more unifying, computational understanding of the cosmos. Given its ambitious
scope, the potential depth of insight into the nature of rules, and its
profound implications, Ruliology demands rigorous exploration and merits
earnest attention in the broader scientific discourse.

It is perhaps worth remarking up front on the similarities to David Deutsch’s
notion of a “constructor” here constructor , because in that case, as in
Wolfram’s approach, one is demonstrating existence through a constructive
(computationally conceived) procedure—both also find some insufficiency in the
orthodox Turing machine model of a universal computer as a model of reality.
What is possible can be constructed _physically_ from some rule (or “task” in
Deutsch’s terminology), and what cannot is impossible (i.e. there is no such
constructor up to the task). Note, also, that as constructive theories, they
have an end-goal in mind (namely, that which is to be constructed), and so
both contain teleological elements.555Indeed, Deutsch has compared his own
approach to cybernetics (which involves steering systems to pre-defined
goals), which he describes as a possible “early avatar”
(https://www.edge.org/conversation/constructor-theory). But while Deutsch
believes he has provided a perfectly non-abstract, physical description of the
world (and, e.g. talks of abstract computers not making sense), he then slips
to mentioning information as if it itself is concrete. It is not, and this is
where Wolfram’s approach has the advantage, since it directly brings in the
additional element that allows for the inclusion of information into the
model. If constructor theory has cybernetics as an early avatar, then 2nd-
order cybernetics is an early avatar of the Wolfram model. Indeed, Deutsch’s
claim about materiality being necessary for realization is a tautology since
by “realization” he _means_ within a material system. This ignores the fact
that there is clearly an information template beyond that realization which is
what such realizations in matter are realizing. There is quite simply no
getting around the fact that if one adopts an information-based ontology, then
one is stepping somewhat outside of orthodox materialism. The key idea of
constructor theory is, then, simply that the focus of fundamental theory
should be which transformations of some medium or substrate into another such
state can be caused to occur, as well, by implication, as those which cannot
be so caused. Given substrate independence, the focus becomes the
_transformations_ themselves as the ontological core of the theory. The
precise nature of this, essentially, modal structure consisting of
counterfactuals has yet to have been adequately nailed down, since while the
transformations themselves are always grounded in some physical substrate, the
counterfactuals, as non-actual by definition, are clearly not (though see
marlettobook  for a discussion of some of the options and problems).

Other related constructivist approaches which lend similar precedence to
processes over substrates include “Assembly Theory,” “Process Theories”, and
“Intuitionistic Physics”. The first of these cronin , focused on the detection
of life, is based on _rules of assembly_ which take into account the number of
independent parts and their connections, such that as the number increases,
the need for memory increases which enables the reconstruction of the whole
from locally stored rules. Process theories are founded in the framework of
category theory and seek to formalize physical operations in diagrammatic
terms in which the diagrams (representing morphisms between objects) are
expressed as objects and transformations within an appropriate monoidal
category cqm1 ; coeckevol ; zx0 ; zx1 ; zx2 ; zx3 . The third approach
(mentioned above) seeks a formalization of physical observations and
measurements based on intuitionistic logic, rather than classical logic, to
escape, for one, the fact that a physics based on real numbers will face the
problem that we will never be able to grasp them, requiring as they do
infinite Shannon information to specify their non-repeating decimal expansions
gisin ; topos1  (note that intuitionistic mathematics involves a temporal,
step-wise process, rather than an eternal, Platonic structure, and this will
be important for our later claims about the essential limitations of the
Ruliad _qua_ fundamental theory).

The substrate-independent approach described above, in constructor theory, is
more or less what Einstein once called the “principle theory” method (see,
e.g. einstein ). Rather than dealing with what things are made of, in terms of
composition, the method looks at the higher-level principles that any and all
things must obey, regardless of their physical constitution.666But note,
again, that on Deutsch’s own interpretation, _some_ physical substrate or
other is required. The substrate would ground particular instances, allowing
for task realization. This transcends particular physical theories and
provides a theory of theories: a meta-theory. Wolfram’s approach shares this
feature of being a theory of theories—one might call it a theory of _all_
theories. Among other things, such a meta-theory bears the burden of having to
explain how physical notions of space, time, matter, laws, and observers
arise, which we identify as the core elements of physical theories. Wheeler
recognized this challenge in the 1970s and coined the term “pregeometry” in
part to address these very issues777A recent formalization of pregeometry
based on homotopy type theory, as well as one based on pre-quantum structures
from noncommutative operator algebras can be found in these works: cpost ;
opalg ; paper2 ; pregeo ; nfold . These studies borrow from ongoing advances
at the foundations of mathematics hott1 ; hott2 ; hott3 ; hott4  as well as
applications of higher category theory to physics baez ; sch1 ; sch2 ; sch3 .
. Note, also, that in delving to this deeper level, one can evade the usual
problems with treating quantum mechanics as a universal theory, in which one
can model not just microscopic systems, but also the very agents using quantum
mechanics frauch18 . In the Wolfram case at least, quantum mechanics is a
feature of the structure itself (the so-called multiway description), a
consequence, rather than the structure being explicitly constructed to capture
quantum features from the outset—the same can be said of the other theories
that fall out of the structure without being inserted _ad hoc_.

In this paper we attempt to explain the Wolfram model, focusing more on
conceptual issues than formal details, and aim to bring out the role of
observers as it appears in the model, showing how it is essential for making
sense of standard physics as well as mathematics. The relationship between the
Ruliad and the observer is also explicated. We use the basic idea of second-
order cybernetics to elucidate a deeper understanding of how the Wolfram model
includes the observer, explaining the very generation of the world we see as a
kind of observer-selection effect, much like a reference frame in relativistic
theories. This allows us, moreover, to provide an account of the nature of
“physical laws” as sampling-invariance. However, the Wolfram model also lets
us see in a new light a fundamental limitation of trying to gain fundamental
knowledge of the world from the standpoint of the observer doing the
modelling. Let us begin with an account of the basic elements of the Wolfram
model.

##  2 The Wolfram Model as an Abstract Rewriting System and The Concept of the
Ruliad

> “When you come to a fork in the road, take it!”
> Yogi Berra

This section provides a literature overview of the basic abstract rewriting
constructions that constitute the Wolfram model, with a particular emphasis on
their non-deterministic aspects, as captured via multiway systems.888A more
detailed version of this material can be found in w1 ; ruliad ; mcomp ; zx1 ,
from which portions of the text in this section are taken (as indicated by
specific citations). We discuss the concept of the Ruliad, representing the
entangled limit of everything that is computationally possible, which emerges
as a key theoretical construct associated to the Wolfram model ruliad ; swtm .
We begin with a preliminary description of the Wolfram model in terms of
diagrammatic rewriting rules acting on hypergraphs. The Wolfram model
represents a discrete framework that posits structures such as continuous
spacetime geometries which may potentially emerge from large-scale limits of
the underlying discrete structures (nks ; w1 ; zx1 ; pregeo ; graphc ).
Furthermore, the evolution of these structures is dictated by various forms of
rewriting rules, such as those based on graphs, hypergraphs, or strings999See
ggrammar ; rewrite  for a background overview on rewriting systems; and arity1
; arity2 ; arity3  for an overview on hypergraph algebras.. To illustrate
this, a Wolfram model hypergraph can be represented abstractly as a finite
collection of ordered or unordered relations (hyperedges) between labelled
nodes, as defined below101010For more detail, see zx1 , section 2, from which
comes some part of the material that follows, including Definitions 1 and 2.
and shown in Figure 1:

######  Definition 1.

A Wolfram Model Hypergraph $H=(V,E)$ is characterized by a finite set of
hyperedges $E$ that belong to the non-empty subset of the power set of $V$,
i.e., $E\subset\mathcal{P}(V)\backslash\\{\emptyset\\}$. A hyperedge in $E$ is
an unordered collection of nodes from the vertex set $V$. Hyperedges in the
Wolfram model, can be both, either directed or undirected.

Figure 1: An example of a Wolfram model hypergraph with directed hyperedges:
$\\{\\{1,2,3\\},\\{2,4,5\\},\\{4,6,7\\}\\}$.

One can then define the dynamics of a Wolfram model system in terms of
hypergraph rewriting rules as follows:

######  Definition 2.

A ‘Rewriting Rule,’ denoted as $R$, for a spatial hypergraph $H=(V,E)$, is an
abstract rewriting rule expressed in the form
$\mathrm{H}_{1}\rightarrow\mathrm{H}_{2}$. In this rule, a subhypergraph that
matches the pattern $H_{1}$ is replaced by a subhypergraph that matches the
pattern $H_{2}$.

######  Definition 3.

A Wolfram model is an abstract rewriting system founded on the principles
outlined in definitions 1 and 2. It’s worth noting that Wolfram models are not
solely limited to hypergraph rewriting systems; they encompass a range of
other rewriting systems, including but not limited to string rewriting
systems, term rewriting systems (TRS), (hyper)graph rewriting systems, and
cellular automata.

Every rewriting rule in this context can be formally mapped to a set-
substitution system, where a specific subset of ordered (unordered) relations
that matches a given pattern is replaced with another distinct subset of
ordered (unordered) relations that also corresponds to a particular pattern,
as shown in Figure 2.

Figure 2: A hypergraph transformation rule corresponding to the set-
substitution system
$\\{\\{1,2\\},\\{2,3\\}\\}\rightarrow\\{\\{4,1\\},\\{4,1\\},\\{4,3\\},\\{2,4\\}\\}$.

It is worth noting that the sequence in which transformation rules are applied
is generally not predetermined. Even in the simplest scenario where the rule
is applied to every matching and distinct subhypergraph (see Figures 3 and 4),
the initial selection of which subhypergraph to transform first remains open-
ended. This multiplicity of choices typically leads to different, non-
equivalent sequences of evolving hypergraphs. Hence, the evolution of a given
spatial hypergraph is inherently non-deterministic due to the absence of a
fixed updating order (or rather the possibility of multiple possible updating
orders). Therefore, we can treat the Wolfram model as a non-deterministic
abstract rewriting system.

Figure 3: The results of the first 10 steps in the evolution history of the
set substitution system
$\\{\\{1,2\\},\\{2,3\\}\\}\rightarrow\\{\\{4,1\\},\\{4,1\\},\\{4,3\\},\\{2,4\\}\\}$,
starting from a double self-loop initial condition
$\\{\\{1,1\\},\\{1,1\\}\\}$. Figure 4:  The result after 19 steps of evolution
of the set substitution system
$\\{\\{1,2\\},\\{2,3\\}\\}\rightarrow\\{\\{4,1\\},\\{4,1\\},\\{4,3\\},\\{2,4\\}\\}$,
starting from a double self-loop initial condition
$\\{\\{1,1\\},\\{1,1\\}\\}$.

Furthermore, within the conventional computational paradigm, systems typically
evolve through a series of sequential steps by applying specific rules.
However, in the case of Wolfram models (for certain rules), determinism is not
inherent. Multiple choices of substitutions are possible, resulting in diverse
outcomes. Usually, we select one possibility (e.g. from the possibility space
mentioned earlier) and disregard the others (as ways the system _might_ have
evolved), but the concept of a multiway system allows for simultaneous
exploration of all potential choices. A key idea is to consider all those
possible threads of history—and to represent these in a single object that we
call a multiway graph.111111In some ways this is similar to the universal
wave-function of Everettian quantum mechanics wal2 , though it sits at a far
lower-level of structure. Indeed, it sits at what might be called a “sub-
structural” level (see paper2  for more on this, including the idea of a
‘structureless structure’ from which structure is generated). Consider as an
example (Figure 5) a system defined by the string rewrite rules: $A\rightarrow
BBB,BB\rightarrow A$. Starting from $A$, the next state has to be $BBB$. But
now there are two possible ways to apply the rules, one generating $AB$ and
the other $BA$ (thus forming a fork in the graph). And if we trace both
possibilities we get what we call a multiway system—whose behavior we can
represent using a multiway graph. And it’s not really difficult to construct
multiway system models. There are multiway Turing machines. There are multiway
systems based on rewriting not only strings, but also trees, graphs or
hypergraphs. There are also multiway systems based on numbers. And all kind of
multiway systems. Combinatorially, a multiway system is simply a directed,
acyclic graph of states, determined by abstract rewriting rules that
inductively generate a (potentially infinite) multiway evolution graph,
together with a partial order on the rewrite rule applications, determined by
their causal structure.

Figure 5: The multiway evolution graph corresponding to the first 7 steps in
the non-deterministic evolution history of the string rewrite rules:
$A\rightarrow BBB,BB\rightarrow A$ (cf. (pregeo, , p. 8)).

Now instead of looking at all possible ways a given rule can update these
rewrite systems, imagine the structure of spaces created by applying all
possible rules. Instead of just forming a multiway graph in which we do all
possible updates with a given rule, we form a rulial multiway graph in which
we follow not only all possible updates, but also all possible rules (an
illustrative figure of rulial multiway graphs for Turing machines with various
numbers of states and colors $\\{s,k\\}$ is shown in Figure 6) ruliad ; swtm .
This construction allows us to think about the notion of rulial space, i.e.
the space of all possible rewriting rules of a given signature. By applying
all possible rules in all possible updates we get what we call the Ruliad the
result of following all possible computational rules in all possible ways (a
schematic depiction of a finite approximation of the Ruliad is shown in Figure
7).121212Here we see another crucial difference to other ensemble theories,
such as the Everettian multiverse, which means that such approaches will be
automatically subsumed in the Ruliad, as the exhaustive application of just
one rule (or category of rules). The Ruliad is instead the ultimate ensemble
theory, or the ultimate multiverse. It’s the ultimate limit of all rulial
multiway systems. And as such, it traces out the entangled consequences of
progressively applying all possible computational rules. The concept here is
to use not just all rules of a given form, but all possible rules. And to
apply these rules to all possible initial conditions. And to run the rules for
an infinite number of steps. Essentially, the Ruliad involves taking the
infinite limits of all possible rules, all possible initial conditions and all
possible steps131313A category theoretic description of the limiting rulial
multiway system in terms of infinity-categories can be found in pregeo ..
Consequently, the Ruliad is in effect a representation of all possible
computations. A conceptual definition of the Ruliad is given below:

######  Definition 4.

_Ruliad_ : A meta-structural domain that encompasses every possible rule-based
system, or computational eventuality, that can describe any universe or
mathematical structure. It acts as a theoretical space wherein the boundaries
between map and territory blur, pushing beyond mere perception and functioning
as the ground for the possibility of multi-computation. Within the Ruliad,
every conceivable physical and mathematical system can be situated, but their
accessibility or meaningfulness is determined by the specific observer-related
frames or constructs. The Ruliad is thus a pre-physical framework, and its
utilization in physics is to pinpoint the exact rule-based system that
corresponds to our observed reality.

A computational definition is as follows:

Let $\mathcal{R}$ be the space of all possible computational (or rewriting)
rules. We refer to $\mathcal{R}$ as the “Rulial space”. Consider
$r\subseteq\mathcal{R}$. $r$ may represent either a single rule or a
collection of rules capable of generating a computational universe $U_{r}$
(the outcome of all possible computations following a given rule set). The
Ruliad, denoted by $\mathbf{R}$, is the collection of all computational
universes. That is, $\mathbf{R}=\left\\{U_{r}\mid r\in\mathcal{R}\right\\}$.

Furthermore,

  1. 1.

An observer $O$ with a frame or construct $F_{O}$ can interpret or access a
subset of $\mathcal{R}$ based on their specific observational constraints. The
observability of any $U_{r}$ is contingent on $F_{O}$.

  2. 2.

The entirety of $\mathbf{R}$ serves as the foundational ground for
multicomputation, transcending the dichotomy between representation and
reality.

Ruliology, intricately intersects with this multicomputational paradigm. This
nexus between Ruliology and multicomputation sets a fresh foundation for
understanding the vastness and versatility of rule space, where different
rules can lead to multifaceted and complex behaviors. Thus, one can contend
that the Wolfram model encompasses a novel paradigm that extends beyond
traditional computation mcomp . This is what Stephen Wolfram calls the
multicomputational paradigm mcomp . It not only traverses the boundaries of
physics but also paves the way for a foundational and versatile methodology
for crafting models in theoretical science. Historically, three paradigms have
dominated theoretical science: mathematical equations, which rely on formulas
to describe phenomena; mechanistic models, which provide detailed, step-by-
step explanations of how systems function, likened to “machines” with distinct
parts and processes; and computational models, which view systems as
computational entities, allowing for the definition of rules and initial
conditions, and then observing the resulting behaviors. However, the
multicomputational paradigm goes a step further. It’s not just about analyzing
specific historical paths but delving into the evolution of all conceivable
histories, epitomized by the Ruliad. In many instances, it may not offer
insights into specific histories. Instead, what it will describe is what an
observer sampling the whole multicomputational process will perceive. This
pivotal intersection of Ruliology and multicomputation is where our
exploration now focuses.

######  Definition 5.

_Multicomputation_ , also known as the multicomputational paradigm, is a
generalization of the traditional computational paradigm to encompass multiple
computational histories or threads of time. In the standard computational
approach, time progresses in a linear fashion. This means that the next state
of a system is computed successively from its previous state. In contrast,
multicomputation allows for every possible path of computation to proceed
through distinct, interwoven threads of time. Instead of a single linear
progression, there are thus multiple threads of computational time that can be
explored. In essence, multicomputation expands the scope of computational
exploration by considering all possible computations simultaneously, rather
than just one at a time.

The following provides a characterization of how computational steps are
executed in a multicomputational paradigm:

Let $S$ be a system defined by a set of computational rules $R$ and initial
conditions $I$. In the traditional computational paradigm, the evolution of
$S$ is represented by a sequence of states $s_{1},s_{2},\ldots,s_{n}$ such
that each state $s_{i}$ is derived from $s_{i-1}$ using the rules $R$. In the
multicomputational paradigm, the evolution of $S$ is represented by a network
$T$ of states, where each node represents a state of $S$ and each branch
represents a possible path of computational evolution based on rules $R$. Each
node can have multiple child nodes, representing different possible next
states. Furthermore, state equivalences between nodes in different branches
allows for intersections of different evolution paths. In other words, for
each state $s_{i}$ in $T$, there exists a set of states $C\left(s_{i}\right)$
such that for each $s_{j}$ in $C\left(s_{i}\right),s_{j}$ is a possible next
state of $s_{i}$ following rules $R$. The network $T$ thus captures all
possible computational trajectories of $S$ starting from initial conditions
$I$.

######  Definition 6.

_Ruliology_ : Derived from the term “rule,” is the systematic study and
exploration of computational rules and their myriad manifestations within
computational systems. It delves into the intricacies of rule space, examining
how diverse collections of rules can give rise to complex behaviors and
structures. Ruliology transcends traditional computational boundaries, aiming
to comprehend the foundational principles behind all possible computations,
and seeking to understand how distinct collections of rules can generate
entire universes of computation. At its core, Ruliology is an attempt to map
out and understand the vast, multifaceted landscape of the Ruliad, where every
conceivable rule is executed in every possible way.

In terms of the Ruliad, Ruliology as the study of computational rules and
universes can be characterized as follows:

Let $R$ be the collection of all possible computational rules, $S$ be the
collection of all possible states, and $F:R\times S\rightarrow\mathcal{P}(S)$
be a map denoting the evolution or transformation of states dictated by a rule
from $R$ upon a state from $S$. The Ruliad $\mathbf{R}$ (defined above)
includes all such computational evolutions. Ruliology is then the study of the
properties, structure, and implications of the Ruliad $\mathbf{R}$, as well as
the exploration of individual and collective behaviors arising from elements
of $R$ when acted upon $S$.

Figure 6: Rulial multiway graphs after 3 steps for Turing machines with
various numbers of states and colors $\\{s,k\\}$ (Adapted from swtm ). Figure
7: A schematic figure of the Ruliad (Adapted from ruliad ). The image
presented can be considered as a rough, finite approximation of the Ruliad.
The complete Ruliad encompasses the exploration of infinite limits across all
conceivable rules, initial conditions, and steps. Here the nodes and edges are
not specific to any single entity. Nodes can represent various entities such
as hypergraphs, strings, or states, while the edges signify the myriad
potential causal connections between those entities.

##  3 Observers, Sampling and the Physical World

> O God! I could be bounded in a nutshell, and count myself a King of infinite
> space… William Shakespeare, _Hamlet_ , II, 2

A potentially serious stumbling block with the Ruliad idea as it stands is
what we might call the “realization problem”: how does an abstract rule get
turned into physical reality? How do we end up with a particular history, to
use the previous section’s terminology? If this reality is the result of the
computation of rules, then what is _doing_ the computation? Surely a computer
of some kind? But then if this is a fundamental theory should this computer
not itself be part of the Ruliad? We find ourselves in this way in a loop
which cannot possibly be a virtuous circle of reasoning. It is more akin to
that famous adventurer Baron von Munchausen rescuing himself and his horse
from a quagmire by lifting himself up by his own hair. We can expect something
like this problem to face any pre-geometry-type proposal that intends to dig
beneath the spatiotemporal world populated with matter to something more
abstract lying beneath—in several quantum gravity proposals the task of
getting the world we are acquainted with, with its description in terms of
fields on differentiable manifolds, from a deeper discrete theory, is known as
the “reconstruction problem”. Part of the problem is that the deeper theories
do not involve things spatially located, evolving dynamically, but an abstract
and far more primitive structure, often based on more relational concepts such
as graphs and networks. It is, of course, a general and well-known problem to
explain how we move from abstract formalism to physical reality. Usually, we
start from the physical reality as a foundation, and then develop an abstract
representation of it. In the case of pre-geometrical approaches (such as the
Wolfram model), one makes no initial theoretical assumptions about the nature
of physical reality, but starts instead from an abstract domain, with the hope
of then recovering the physical aspects from this.

A related problem, of moving from abstract to concrete, is well expressed by
John Wheeler (wheegrav, , p. 1208):

> Paper in white the floor of the room, and rule it off in one-foot squares.
> Down on one’s hands and knees, write in the first square a set of equations
> conceived as able to govern the physics of the universe. Think more
> overnight. Next day put a better set of equations into square two. Invite
> one’s most respected colleagues to contribute to other squares. At the end
> of these labors, one has worked oneself out into the doorway. Stand up, look
> back on all those equations, some perhaps more hopeful than others, raise
> one’s finger commandingly, and give the order “Fly!” Not one of those
> equations will put on wings, take off, or fly. Yet the universe “flies”.

A well-known related sentiment was expressed through Stephen Hawking’s
question “What breathes fire into the equations?” in his book, _A Brief
History of Time_ SH . In other words, what make an (abstract) equation or
generalization (which ‘oversees’ a set of possibilities) a physical reality?
Hawking elaborates:

> Even if there is only one possible unified theory, it is just a set of rules
> and equations. What is it that breathes fire into the equations and makes a
> universe for them to describe? The usual approach of science of constructing
> a mathematical model cannot answer the questions of why there should be a
> universe for the model to describe. Why does the universe go to all the
> bother of existing?

This is an issue concerning the metaphysics of the laws of nature. We can give
much the same response for the Ruliad as Wheeler did here: observership, or
rather participation, provides the necessary engine that powers the creation
of the physical universe. Hence, the Ruliad will need to include a “theory of
the observer” such that “the universe as we know it” seen from the vantage
point of “observers like us” realizes observable physical attributes of the
universe. However, it would be a mistake to view the physical universe as
_unique_ and _absolute_. Moreover, any computation is a result of the
perspective of a computationally-bounded observer, rather than a fundamental
feature of the universe itself: we are now in the realm of epistemology (i.e.
description or representation) rather than ontology (how things are in a
fundamental sense). The observer acts as a kind of transducer for the Ruliad,
converting the abstract computations into (apparently) physical
form.141414Though ultimately everything (observer and observed) is supposed to
remain part of the Ruliad of course, and so will remain abstract when
conceived from a third-person perspective. However, if given a complete
treatment, then the notion of the Ruliad is also representation via an
observer.

There is here an interesting philosophical relation of the Ruliad to Leibniz’s
system of monads. Leibniz’s _monadology_ , his last attempt to codify his
philosophical system, can certainly rival Wolfram’s Ruliad for all
encompassing majesty, despite its extreme brevity. Each monad is an individual
that reflects the rest of the universe from its own unique point of view. The
parts shape the whole and in turn, the whole back-reacts on the parts.
Likewise, the Ruliad has similarity to _Indra’s Net_ from _The Flower Garland
Sutra_ \- a kind of representation of a totality in terms of bejeweled
vertices which encode the whole. Each is a vista of the whole. Every possible
view is present in the whole. It is interesting to see how this basic idea, in
which a totality is decomposed into an interdependent parts, repeats.151515See
atrick  for a treatment of such a decompositional metaphysical position
(decompositional dual-aspect monism) as elucidated in several case studies
from physics, of which the Wolfram model appears to be another convincing
instance.

While monads collectively provide all possible perspectives of a world, as
tiny independent mirrors (or points of view), Wolfram’s Ruliad deals with all
possible rules applied to some initial collection of abstract relations.
However, there is also a sense in which the monads are carrying out a pre-set
program (or entelechy), coordinated with all other monads, in a pre-
established and divinely choreographed dance determined to generate (i.e.
construct) the best of all possible worlds. While there are the well-known
principles of sufficient reason and identity of indiscernibles providing basic
constraints on this construction, the principles themselves do not directly
determine what is constructed. Rather, they inform the composition of the
monads into complex structures which is then carried out through the pre-
established harmony. A major reason for the introduction of pre-established
harmony was to explain the mind-body (or soul-body) correlations. For Leibniz
there was no causal link and the correlation stemmed from the common cause in
which both were set on their way like a pair perfectly synchronized watches.
Interestingly, as we will develop further in another paper paper2 , Wolfram’s
model has a remarkably similar explanation for the correspondence of the world
to the mind in that they both emerge from the same initial rules for
construction and emerge in parallel with the mind (or observer) simply
sampling the world and providing a perspective, much like a monad, where
different observers represent the whole universe from different points of
view. Likewise, one can find a similar generation of variety in the Wolfram
model through this dislocation of a single, unified structure into many of
points of view.

Of course, Leibniz’s theory, as it stands, is not one of much practical value
in terms of showing how our present theories and phenomena can be
_constructed_. The approach of the Wolfram model, involving hypergraph
rewriting systems, places the ontological weight on the very rules of
construction themselves. By contrast with Leibniz’s “God as architect” (as he
puts it in S.89 of his _Monadology_), here the metaphor is better expressed,
following Chaitin chaitin , as “God as programmer,” though here employing a
multiway approach rather than a single-track, Turing machine approach. A more
crucial distinction, related to the constructive approach, is that the physics
(and mathematics) emerge from the interplay of computationally-bounded,
embedded systems (observers) with the structure in which they are embedded
(and therefore sampling), namely the Ruliad.

What an observer ultimately does is to take in an input from a large set, and
return an output from a much smaller set, thus acting as a kind of idempotent
filter. It’s a concept that’s appeared in many fields under many different
names. It can be called a contractive mapping, a reduction to canonical form,
a classifier, a forgetful functor, lossy compression, projection,
renormalization group transformation, and so on. It’s what’s fundamentally
going on whenever we use a sensor or a measuring device, or for that matter,
our human senses: we extract statistics, fit to models, and describe things
symbolically. As a basic physical example, consider a gas pressure sensor
based on a piston. Within the gas, individual molecules move around in
complicated and seemingly random ways, hitting the piston in all kinds of
configurations. But the piston “reduces out” all those details, responding
just to the aggregate force of all the molecules, the same one of which can be
realised in potentially infinitely many ways. The main point is that we can
describe what’s going on more formally by saying that “observations by the
piston” identify all the different detailed configurations of molecules,
preserving only information about their aggregate force, forgetting the finer
details.

This same idea can then be given slightly different interpretations, revealing
how observers influence various branches of science. In statistical physics,
for example, observers have, as just described, the effect of averaging over
many particles or other degrees of freedom sw2law . In General Relativity,
they are averaging over spacetime regions, and forgetting those details having
to do with coordinate transformations. In quantum physics, they are basically
averaging over many quantum histories. In mathematics the “same” statements
are stated differently in terms of underlying axioms (see w3 ). Gauge theories
can be understood in the same way: the equivalence classes will in this case
be generated by the gauge transformations which will be identified relative to
an observer (though perhaps some other observer could view absolute structures
such as the individual gauge potentials, much as a skilled musician with
absolute pitch can hear differences that most others identify). In high energy
physics, black holes in various dimensions have an associated thermodynamic
description, such that the physical charges of the black hole or black ring
depend on whether the object being is viewed from a 4D perspective or a 5D
perspective 5d1 ; 5d2 . Further afield, in economics, the focus might be on
certain indices generated by the behaviours of a country’s people. Even in
linguistics we have identifications (between systems that differ in the
details) given by the equivalence relation defining concepts, such as what
counts as a chair, coat hook, or cabbage. This can then be applied to all
scientific areas (and beyond) in which observers are involved.161616Note that
we are able to see a clear explanation of the so-called “unreasonable
effectiveness of mathematics” here wigner . The mathematical and physical
structure emerge from the selfsame source, namely ruliadic sampling. Since
they are constrained in the same way, a specific observer’s sampling system
will pick up correlations between the systems and properties it generates, and
the way they are encoded in mathematics.

Let us now consider some further conceptual implications of this overall
structure or observers sampling the Ruliad, along with the notion of
updating/rewrite rules. We start with the status of the approach _vis-à-vis_
determinism and modality.

##  4 Computation, Determinism and Free Will

The question of determinism versus indeterminism is not so clear-cut in the
Ruliad model and intersects with the issue of descriptions from the inside
versus outside [i.e. computationally unbounded versus computationally bounded
respectively]. One reason is that the inclusion of quantum mechanics into this
picture is achieved through a notion of a branching (i.e. multiway), rather
than linear, structure connecting the states in a process. This is supposed to
represent the multiplicity of possible paths that quantum, but not classical,
mechanics entails. In much the same way that the universal wave function,
while itself deterministic (evolving according to the linear Schrodinger
equation) nonetheless contains a kind of _local_ indeterminism if one follows
specific paths through the space defined by the universal wave-
function.171717We have in mind the Deutsch-Wallace wal  approach to the
interpretation of probabilities in quantum mechanics according to the Everett
interpretation in which one also has to square an ultimately deterministic,
branching process with the apparent indeterminism in measurement results.
Indeed, there is no reason why one might not adopt the self-same decision-
theoretic approach in the Wolfram model in which ‘things occur’ (i.e.
outcomes) only relative to the embedded observers. The idea would, in this
case, be to view the probabilities in terms of rational decision under
uncertainty about ones location in the Ruliad. In other words, our answer to
the question “Is this theory deterministic or indeterministic?” hinges on
whether we are viewing things from an embedded perspective or from a God’s eye
perspective.

However, to view this branching itself as parallel to the indeterminism of
quantum mechanics is to deny the possibility of classical, deterministic
physics. And, of course, the Wolfram model should contain both classical and
quantum theories, according to its role as a theory of all theories. Wolfram
has a way of explaining this, of course, by pointing to a coarse-graining
effect mentioned in the previous section. The explanation of definite
happenings [phenomena] despite the multiplicity of paths then comes about
through the observer-participator as embedded in the multiway system. However,
as with many-worlds interpretations, from the perspective of the totality (the
Ruliad), everything that can happen (from the point of view of rules) does
happen. However, the approach here goes beyond many-worlds—or is perhaps more
in keeping with Hugh Everett’s original ideas—since it incorporates the
observer in the model itself (something we explore more in the next section).

Given the branched-system understanding of quantum mechanics, it is clear that
the model can support a grounding of counterfactuals along the same lines as
constructor theory, in which the Everettian approach is adopted (at least for
counterfactuals that occur in some branch). One of the problems faced in that
approach is precisely that there are modal claims (e.g. those involving the
very laws of physics themselves) that do _not_ occur in a branch of the
multiverse. In this sense, the Ruliad has more modal breadth, since it
encompasses entire theoretical structures.

Does this have anything to tell us about free will? One might, for example,
take the idea of computational irreducibility, in which one must let a system
evolve to know its evolution, as some kind of free will proxy. However, this
is simply unpredictability, not indeterminism. We find the same behaviour in
chaotic systems, of course, where it is understood that the systems are
perfectly deterministic albeit with agent-relative uncertainty about the
development. It is why we simulate some systems, and in which the best we can
do to know a future state is simply increase the performance power of the
computer cranking through each iteration. So as observers watching a
computationally irreducible process, we have uncertainty about the future, but
the future is not uncertain _simpliciter_. This must be the case if the Ruliad
is seen to exist eternally, so that it is not a growing block structure. Such
a degree of uncertainty does not imply free will. Even if there was pure
randomness, it would not imply free will. Free will is the ability for the
evolution of a system to fork in such a way that an agent has the ability to
decide which path is chosen. That is, one could have isomorphic histories up
to a point $t$ which diverge thereafter. There seems to be a confusion
occurring between one’s actions not being pre-determined and free will. But
with the Ruliad we have neither of these situations, but simply an epistemic
uncertainty about the future based on embedding an observer in a system with a
kind of temporal gradient that emerges from the hypergraph rewriting process
that the selfsame observer will be part of.

Wheeler famously asked “Why the quantum?” We have an answer here: the Ruliad
(and its included samplers). And if we are then pushed to ask Why the Ruliad?
Then we have an answer there too: there could not _not_ be the Ruliad, since
it is a mathematically necessary object. But we can say a little more here.
The quantum is not only a matter of the multiway picture. It is also a demand
that the observer be included. Without this we have something akin to a space
of pure potentiality, but in which nothing is made concrete or actual. There
are no happenings without the consideration of a frame with respect to which
something happens. This is very similar to the way in which there is no moon
in quantum mechanics when no one is looking: there simply is no objective
_way_ the world is in quantum mechanics, and likewise not in the Wolfram model
either. Rather, there are all ways, which implies no way. Hence the need to
introduce something like a Wheelerian observer-participator to select one such
way the world can be, though with uncertainty as to which way that is. This
notion of including the observer in so central a way suggests that the Wolfram
model would benefit from ideas originating in 2nd-order cybernetics. Indeed,
the Wolfram model might be a fine example of a naturally 2nd-order cybernetic
system.

##  5 The Algorithmic Nature of Observers

Circling back to the dual relation between computation and measurement of an
observable, one may think of measurement as an inference process under
conditions of uncertainty. In other words, the process of observation (at
least in the sense of making measurements of physical quantities) is the act
of inferring approximate or coarse-grained causes of the outcomes that
computations (around the observer) generate. Given the observer’s own
computational boundedness, it has to base these estimates on limited samples
and restrict to causes that provide description only up to those levels of
complexity that its own inferential engine can handle. This is where making
equivalences and coarse-graining states of high complexity become relevant for
any observer theory. This suggests an algorithmic nature of the observer as an
inference engine within the Ruliad seeking states or computations with lowest
complexity.

Apart from the setting of the Ruliad, a theory of inference engines has been
described in the context of cognitive neuroscience in terms of Karl Friston’s
“Free Energy Principle,” where a cognitive agent seeks to minimize its free
energy either by performing actions upon the world or by changing its
perceptions/representations of the state of the world based on new incoming
data friston . The free energy here is a complexity measure181818See con4  for
an overview of complexity measures related to cognition and consciousness.
whose minimization is associated to minimizing the “surprise” or uncertainty
in the agent’s representation of the world around it. More generally, for
cognitive agents (both, biological and artificial), this minimization is
achieved algorithmically using a hierarchical inference scheme based on
feedback loops involving predictions and errors concerning the states of the
world in comparison to the agent’s own prior expectation con2 ; morpho .

Coming back to the Ruliad, the idea of an observer as an inference engine may
be abstracted as a theory of sampling and measurement of low-complexity (or at
least comparable to the observer itself) states within the Ruliad. Any
complexity minimization principle akin to the free energy principle is in fact
a second-order cybernetic construct. Presumably, this has to be included as a
meta-rule upon the Ruliad.

In contrast, Roger Penrose has famously defended the view that human
consciousness is non-algorithmic penrose  (see also con1 ; con3 ). Prima
facie, if we are treating observers as an emergent feature of the Ruliad, then
we must respond to Penrose’s challenge. If we take Penrose’s view of
consciousness, as developed with Stuart Hameroff hp , then we can see how this
can be accommodated by accommodating the microtubles within the Ruliad, and
having an account of the coherence they exhibit.

Penrose and Hameroff posit that orchestrated objective reduction of the wave
function is associated to proto-consciousness and this is non-computable. When
Wolfram speaks of computation as omnipresent, he refers to a general use of
the term that includes both computations that are reducible as well as
irreducible. It is the irreducible ones that correspond with what Penrose
refers to as non-computable. From the point of view of microtubuli represented
within the Ruliad, they are running some irreducible rules (analogous to CA
rule 30 nks ) which to another observer (within the Ruliad) does not lend
itself to full predictability. When an observer conflates histories of the
multiway, that constitutes a measurement upon the external world. This
measurement itself may be computationally reducible. But the observer also
needs a higher-order computation which determines which measurements to make
and which histories of the multiway it should conflate - that higher-order
process may be irreducible (one may call that meta-cognition). If these
higher-order processes are required for consciousness, then the conscious
observer is not just a program, but a meta-program (and an irreducible one).

Ultimately, what computational irreducibility means is, as the name suggests,
that there is no redundancy in the process that can be eliminated to shorted
or compress the process. This means that there can be no ‘short cuts’ in which
features of the process can be ignored, made equivalent, or in some other such
way utilised to jump to the end. The best one can do is to run the process or
simulate it. Again, one might be able to throw more performance power at at,
but still it must run through step-by-step. In this sense we see that
predictability is bound up with the notion of reducibility, and we have
something like an open future if not quite full-blown free will. We are thus
left with incompleteness, however, which is related to the so-called hard
problem of consciousness. We are giving a model of an observer ‘from the
outside’ as it were. Yet how do we find a place for subjectivity (the inside
view) here?

##  6 Seeing the Ruliad from the Inside: Second-Order Cybernetics

> Space and time, defining everything we cognize by sensuous means, are in
> themselves just forms of our receptivity, categories of our intellect, the
> prism through which we regard the world - or in other words, space and time
> do not represent properties of the world, but just properties of our
> knowledge of the world gained through our sensuous organism. From this it
> follows that the world apart from our knowledge of it, has neither extension
> in space nor existence in time; these are properties which we add to it. (P.
> D. Ouspensky, _Tertium Organum_ , 4)

The notion that the world we experience (the phenomenal world or manifest
reality) is _conditioned_ by our faculties as observers, including the notion
of computational boundedness or limitation, can be traced to Kant’s theory of
the categories. This traces many features that we might naively impute to the
world itself back to features of the observer. A natural question, and one
considered by Kant, is what happens when different observers are considered.
The Wolfram model also involves the idea that different observers might
generate very different descriptions of the Ruliad, and so would discover
different laws in their world. It is, in other words, vital that the specifics
of observers be provided, in order to get a world-description out, and as such
the former is the _sine qua non_ of the latter.

Several examples of such ‘alien’ scenarios were presented in the early
flatland ideas. While Edwin Abott’s approach is the best known, the most
useful for our purposes is Charles Howard Hinton’s, who writes:

> Thus if we make up the appearances which would present themselves to a being
> subject to a limitation or condition, we shall find that this limitation or
> condition, when unrecognized by him, presents itself as a general law of his
> outward world, or as properties and qualities of the objects external to
> him. He will, moreover, find certain operations possible, others impossible,
> and the boundary line between the possible and impossible will depend quite
> as much on the conditions under which he is as on the nature of the
> operations. (hinton, , p. 40)

Our epistemological equipment allows us to generate a kind of screen on which
reality can display itself. But, of course, what is manifest is only a
relative appearance and has much to do with the equipment (including any
necessary factors that enable it to exist in the first place). Thus, the
observer (human or otherwise) acts as a kind of prism, or transducer,
converting a potentially infinite spectrum of data into a finite package
capable of being processed. The prism is a good analogy because without it,
there would be no such phenomena. And had we placed a distinct observer where
the prism is, perhaps a mirror, then we would generate a very different kind
of display of the _same_ region.191919Note also that some of these ‘displays’
might be mutually incompatible leading to the kinds of complementarity one
finds in quantum mechanics, e.g. with the inability to place equipment capable
of both position and momentum measurements. But, to repeat, without some means
of displaying the world, there is nothing there other than a kind of
potentiality to display.

Second-order cybernetics is based on the idea that no science is possible from
a ‘view from nowhere’ in which one can view reality unveiled as it were cyb1 ;
cyb2 ; cyb3 ; vfUU . One has to consider a standpoint, or perspective, or
frame from which the universe is viewed. Without it (i.e. the viewer), there
is no view. The Ruliad as it stands, is abstractly defined as a view from
nowhere: a totality. Wolfram himself speaks explicitly of the Ruliad “viewed
from the outside” (w1, , p. 235). To carry out scientific exploration in the
Ruliad, we must include a system, an observer, capable of sampling the
space.202020We don’t go into any details here, beyond simply noting that the
Wolfram model fits the basic mould of the 2nd-order cybernetic framework. A
future paper will consider the pairing in more details. See vf  for a superb
review of the basic ideas of 2nd-order cybernetics.

Wolfram elsewhere defines the Ruliad as “result of following all possible
computational rules in all possible ways” ruliad ; w3 . This is more in line
with the 2nd-order cybernetics approach, but we must ask: _who_ is following
the results? Who is the observer in this case? And who models that observer?
From the outside, the Ruliad is simply understood as the totality of all
possible computations. However, from the point of view of any of its parts
(which satisfy criteria of computational boundedness and persistence), any
part of the Ruliad, qualified by boundedness and persistence, is potentially
an observer of its complement (within a specified horizon, which would again
depend on its computational boundedness). The computations performed by this
localized observer realize measurements in the universe. Hence, from the
perspective of 2nd-order cybernetics, the object of interest may not be the
Ruliad by itself, but rather the power set (or appropriate categorical
generalization of a power set) of the Ruliad that encapsulates observers, the
observed and their interactions. Teleology and mereology will both be relevant
to this power object (of the Ruliad).

In the past, it was usually possible to do theoretical science without
explicitly discussing the observer. But it turns out that to say anything
about “what happens” requires knowing about the observer. In general, what an
observer does is take the raw complexity of the world, and reduce it in such a
way that conclusions can be made from it. And here one can think of a certain
fundamental duality between computation and observation: the process of
computation has the effect of generating new outcomes; the process of
observation has the effect of reducing outcomes by ‘equivalencing’ different
ones together, as discussed earlier. Computation theory gives us a way to
describe possible processes of computation. And the goal of what we’re calling
“observer theory” is to give us an analogous way to describe possible
processes of observation. Because it turns out that our limitations as
observers are in a sense what gives us many of the most fundamental scientific
laws that we perceive. And it’s really all about the interplay between the
underlying computational irreducibility and our nature as computationally
bounded observers.

The crucial feature of observers seems to be that the observer is always
ultimately some kind of “finite mind” that takes all the complexity of the
world and extracts from it just certain “summary features” that are relevant
to the “decisions” it has to make swcon . Observers like us have two basic
characteristics: first, that they are computationally bounded, and second,
that they are persistent in time. Computational boundedness is essentially the
statement that the region of space observers occupy is limited, i.e. we can’t
expect to “reverse engineer” computationally irreducible processes that are
going on “underneath” swcon .

The Wolfram model takes the perspective that an observer has to be a part of
the underlying multiway system (possibly as a subgraph spread across
branches). In this view, measurement is consequently the process of the
observer conflating parallel threads of multiway history with a single
evolution leading to the illusion of a unique sequential thread of time.
Furthermore, the notion of causal invariance, which can be thought of as being
associated with paths of history that diverge eventually converging again, is
what guarantees a coherent eventual consistency. And since the Ruliad contains
paths corresponding to all possible rules, it’s basically inevitable that it
will contain what’s needed to undo whatever divergence occurs—because of
causal invariance the laws of physics are invariant in any frame of reference
(though to realize the laws one has to set up a system within a given
reference frame). Hence, anything physically observable is going to be in a
subjective setting, and from this subjective setting, when we infer anything
objective, it is relative-objectivism.

So any given observer is interpreting what they see in terms of a description
language, which causes them to attribute certain rules to be “the rules of the
universe”—one has to choose what kind of system one is working with, and it is
almost impossible to state a law without those choices. Once we make those
choices, we are already in a constructive domain. Hence, if one sets up some
particular computational system or mathematical theory there will always be
choices to be made, and our most important feature as observers is that we’re
computationally bounded, i.e. the way we parse the universe involves doing an
amount of computation that’s absolutely tiny compared to all the computation
going on in the universe. We sample only a tiny part of what’s really going on
underneath, and we aggregate many details to get the summary that represents
our perception of the universe. Recall our earlier example of the molecules in
a gas. The molecules bounce around in a complicated pattern that depends on
their detailed properties, but an observer like us doesn’t trace this whole
pattern. Instead, we only observe certain “coarse-grained” features (e.g.
pressure and temperature). In this sense everything then boils down to how an
observer chooses/samples the space in which they are located, so that their
properties are of the essence, which reveals the Wolfram framework as an
already 2nd-order cybernetic system.

##  7 The Limits of Ruliology: The Impossibility of Seeing the Ruliad from the
Outside

> Philosophy is an attempt to express the infinity of the universe in terms of
> the limitations of language.
>
> A. N. Whitehead

If, as Whitehead put it, philosophy is an attempt to express the infinity of
the universe in terms of the limitations of language, Ruliology is likewise an
attempt to express that infinity in the somewhat less limited framework of the
representations of computationally-bounded observers that are embedded within
it. The Wolfram model depends on coarse-graining over paths in order to model
the observed physics. The coarse-graining is, of course, relationally linked
to specific observers (or classes of observers). This introduces a limit,
since any ruliadic properties that we can speak about are of course from the
point of view of a member of such a class of observers. We can model other
observers by changing the properties defining the observer-class, but even
this is itself generated from our own perspective and so will inherit any
associated limitations.

Following on from the two ways of thinking about the Ruliad, from the inside
versus the outside, we can see that the irreducibility is a feature of the
embedded view: it is a feature of the relationship between observers and the
Ruliad of which they are a part. We can link this to the two broad approaches
to the ontology of mathematics, and note the role they play in physics. From
the inside view, in this case, the appropriate ontological picture is that of
constructivism, with intuitionistic logic playing the role, and in someway
paralleling the computational irreducibility that the observers face in their
knowledge claims. But taken as a completed object, where all processes have
been carried out to their infinite limits, then there is of course no
computational irreducibility, because the object is eternally given, and we
view it _sub specie aeternitatis_.

It seems that Wolfram is acutely aware of the necessity to include the
observer in the description itself sw2020 :

> It’s a typical first instinct in thinking about doing science: you imagine
> doing an experiment on a system, but you—as the “observer”—are outside the
> system. Of course if you’re thinking about modeling the whole universe and
> everything in it, this isn’t ultimately a reasonable way to think about
> things. Because the “observer” is inevitably part of the universe, and so
> has to be modeled just like everything
> else.212121https://writings.stephenwolfram.com/2020/04/finally-we-may-have-
> a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/

Wolfram writes that w3 :

> [T]he Ruliad is not just a representation. It’s in some way something lower
> level. It’s the “actual stuff” that everything is made of. And what defines
> our particular experience of physics or of mathematics is the particular
> samples we as observers take of what’s in the
> Ruliad.222222https://www.wolframscience.com/metamathematics/counting-the-
> emes-of-mathematics-and-physics/

This stuff is made of “emes,” which function as the most fundamental layer w3
. It is supposed to transcend the observers, and goes beyond representation.
However, given our discussion of 2nd-order cybernetics, can this be right? How
can we possibly make any statements about reality that do not carry with them
their source from us _qua_ observers? To say it is not just representation
implies that we can somehow step outside of all representation, and step
outside of our position as observers, to see that more lies beyond: _plus
ultra_. In doing so we have stepped outside of ruliology proper, and entered
speculative metaphysics. Rather, as Heinz von Foerster puts it:

> [A] brain is required to write a theory of a brain. From this follows that a
> theory of the brain, that has any aspirations for completeness, has to
> account for the writing of this theory. And even more fascinating, the
> writer of this theory has to account for her or himself. Translated into the
> domain of cybernetics; the cybernetician, by entering his own domain, has to
> account for his or her own activity. (vfUU, , p. 289)

The observer in the Wolfram model must ultimately also be ruliadic if this
theory is to be truly fundamental. Indeed, the necessity for second-order
cybernetics suggests the need for _meta-rules_ upon the Ruliad itself. These
rules when instantiated locally within computationally bounded patches of the
Ruliad operationalize an abstract notion of observers.

In this case, when we speak of ourselves sampling the Ruliad to generate
particular systems of mathematics and physics, we are really speaking about
the Ruliad _self_ -sampling.232323This is also known as endophysics, or the
physics from the inside (see e.g. rosser ). Rössler argues that the world is
always relative to an observer-perspective, so that an “interface” is involved
in which the world appears as a kind of screen to the observer, though not in
any fixed way. Rather, the cut between self and world (or observer and
observed/environment) is variable. In the case of the Observer-Ruliad system
the observer is a kind of bounded foliation of the whole.  In this way we can
compare the role and status of observers in the world to the role of humans in
such religious systems as Sufism. A totality has splintered into many
(relative) perspectives, each with the ability to explore a particular part of
that totality. The perspectives are transducing the ineffable Ruliad (akin to
an _absolute_ unconditioned reality) into something ‘effable.’ The self-
sampling naturally leads to reflexivity and looping elements, linking the
observer and observed. As Kauffman explains:

> In an observing system, what is observed is not distinct from the system
> itself, nor can one make a separation between the observer and the observed.
> These 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 converse with us. We are the models. Map and territory are
> conjoined. (kauf1, , p. 1)

This is not a flaw with such a model, but rather a virtue. It is quite clear
that if we consider some global system [a universe or Ruliad] then we can see
that is is trivially the case that some sub-system [the observer] of that
system cannot observe the whole system. In other words, the system as a whole
is not an observable in the strict sense: there is no operation that we can
envisage to measure it.242424Here we are assuming that the system is closed,
of course, so that there is no external interaction and no sense in which the
system is itself a sub-system of some larger system. In the case in which,
e.g., the system is a simulated universe, then of course one can well imagine
an external observer being able to make appropriate measurements on the
system, though they would then face the same problem in their own universe
(cf. (fields, , 2.1), fields0 ) and the description in question would no
longer be fundamental. That the Wolfram model contains observers and their
viewpoints, and generates physics through sampling [in a consistent loop], is
the most fundamental model one can manage if one does indeed take seriously
the fact that we ourselves must be such observers. It is a virtue for a theory
to describe its limitations, in this case containing its limits as theoretical
outcomes.

Of course, the Ruliad is not something that we could ever directly observe,
and nor is it presented as such. It is an abstract entity that if picturable
in any way would be akin to a kind of hyper-tensorial object. However,
inasmuch as it is abstract, it exists as representation in the mind of an
agent and so inherits the limitations. In this case we cannot quite speak of
the distinction between map and territory blurring, as with perceptions of the
world, because the Ruliad is supposed to go beyond any possible perception. In
this sense it stands more in the position of an unknowable God, and the
evidence we have is more of the form of a transcendental argument, such that
it functions as the _ground_ required for having the kinds of experiences we
do have and for there being an apparently existing universe in the first
place. It is the ground of the possibility of multi-computation.

As we are familiar with from general relativity, although there are many
systems of coordinates that can serve to fix a gauge on the universe, each
providing a foliation of the spacetime that is invariant when completed. There
is also the notion of the quotient space that, in a sense, averages over all
the gauge freedom, spitting out the invariant structure.252525In a similar
way, Wolfram speaks of “bulk” features, in the sense of something like the
quotient object, i.e. that which transcends the baggage brought by observers
and their gauges or frames. Of course, the resulting entity is harder to deal
with from a physical point of view since the gauge (the coordinate frame) are
what allows us to epistemically access the universe. Of course, this also
tells us that the epistemic access is also partly one of construction of what
is observed. The frames are observer-constructs and must be purged in any
consideration of what is the ‘true picture’ of reality. Of course, one can
also simply speak of the frame-relative picture in such a way that so long as
the frame is considered in the evaluation of some physical quantity there is a
kind of observer-invariance of the quantities by virtue of observers being
able to change their own coordinates into new frames.

This issue of attempting to describe a reality beyond our selective,
descriptive capacities is a common problem facing those dealing with apophatic
theology, in which one cannot speak of the thing in question with positive
characteristics because they thereby bound it, and yet the very bounds come
from us. But without such an extremely deep level of probing, we cannot be
said to have a fundamental theory. By focusing on the building rules
themselves we find both physics and mathematics emerging, which is as we
should expect. Along these lines, a complaint that has been levelled against
the Wolfram model described here is that it is incapable of making
predictions. The same complaint was lodged against Eddington’s Fundamental
Theory. But it entirely misses the point, which is that the Ruliad is a home
for physical theories. It is the ground.

A fundamental theory, that is this fundamental, cannot possibly be expected to
make direct predictions of the sort that the critics clearly desire. But what
it can do is locate them in a web of theories, and moreover it can suggest
entirely new kinds of theory that would then themselves make predictions when
properly worked out in the manner appropriate for less fundamental approaches.
The task of physics, indeed, is to figure out where in the Ruliad we are
located. In this sense physics (and mathematics) amount to the task of a
librarian working in Borges’ Total Library. Every possible book is contained
within that library, much as every possible theory is contained in the Ruliad.
While all possible books are therein, one needs the right indexing system to
locate the correct book, so as not to grab some book of gibberish. The
observer is the linchpin that connects the Ruliad with scientific theories,
since it is the locus of indexing. Ruliology is not then, a replacement of
physics, but a way of making sense of it. Moreover, what might appear to be
physically nonsensical sectors of rulial space to us, might be perfectly
experienciable (as a quite different kind of universe) to other kinds of
observers. What we have described is, then, not a theory of physics in the
ordinary at all. It is a pre-physical framework for any possible theory of
physics and should not be analysed (or critiqued) in the same terms as
orthodox physical theories.

### Acknowledgments

We thank Stephen Wolfram for engaging discussions on the Ruliad. This work was
supported by the following grants: The Foundational Questions Institute and
Fetzer-Franklin Fund, a donor advised fund of Silicon Valley Community
Foundation [FQXi-RFP-1817]; The John Templeton Foundation [Grant ID 62106];
and the Australian Research Council [Grant DP210100919].

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