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© 1998 by Christopher Michael Langan The real universe has always been
theoretically treated as an object, and specifically as the
composite type of object known as a set. But an object or set
exists in space and time, and reality does not. Because the real universe
by definition contains all that is real, there is no "external
reality" (or space, or time) in which it can exist or have been
"created". We can talk about lesser regions of the real universe
in such a light, but not about the real universe as a whole. Nor, for
identical reasons, can we think of the universe as the sum of its parts,
for these parts exist solely within a spacetime manifold identified with
the whole and cannot explain the manifold itself. This rules out
pluralistic explanations of reality, forcing us to seek an explanation at
once monic (because nonpluralistic) and holistic (because
the basic conditions for existence are embodied in the manifold, which
equals the whole). Obviously, the first step towards such an explanation
is to bring monism and holism into coincidence. When theorizing about an all-inclusive
reality, the first and most important principle is containment,
which simply tells us what we should and should not be considering.
Containment principles, already well known in cosmology, generally take
the form of tautologies; e.g., "The physical universe contains all
and only that which is physical." The predicate "physical",
like all predicates, here corresponds to a structured set, "the
physical universe" (because the universe has structure and contains
objects, it is a structured set). But this usage of tautology is
somewhat loose, for it technically amounts to a predicate-logical
equivalent of propositional tautology called autology, meaning
self-description. Specifically, the predicate physical is being
defined on topological containment in the physical universe, which
is tacitly defined on and descriptively contained in the predicate physical,
so that the self-definition of "physical" is a two-step
operation involving both topological and descriptive containment. While
this principle, which we might regard as a statement of "physicalism",
is often confused with materialism on the grounds that
"physical" equals "material", the material may in fact
be only a part of what makes up the physical. Similarly, the physical may
only be a part of what makes up the real. Because the content of reality
is a matter of science as opposed to mere semantics, this issue can be
resolved only by rational or empirical evidence, not by assumption alone. Can a containment
principle for the real universe be formulated by analogy with that
just given for the physical universe? Let's try it: "The real
universe contains all and only that which is real." Again, we have a
tautology, or more accurately an autology, which defines the real
on inclusion in the real universe, which is itself defined on the
predicate real. This reflects semantic duality, a logical
equation of predication and inclusion whereby perceiving or semantically predicating
an attribute of an object amounts to perceiving or predicating the
object's topological inclusion in the set or space dualistically
corresponding to the predicate. According to semantic duality, the
predication of the attribute real on the real universe from within
the real universe makes reality a self-defining predicate, which is
analogous to a self-including set. An all-inclusive set, which is
by definition self-inclusive as well, is called "the set of all
sets". Because it is all-descriptive as well as
self-descriptive, the reality predicate corresponds to the set of all
sets. And because the self-definition of reality involves both descriptive
and topological containment, it is a two-stage hybrid of universal
autology and the set of all sets. Now for a brief
word on sets. Mathematicians view set theory as fundamental.
Anything can be considered an object, even a space or a process, and
wherever there are objects, there is a set to contain them. This
"something" may be a relation, a space or an algebraic system,
but it is also a set; its relational, spatial or algebraic structure
simply makes it a structured set. So mathematicians view
sets, broadly including null, singleton, finite and infinite sets, as
fundamental objects basic to meaningful descriptions of reality.
It follows that reality itself should be a set…in fact, the
largest set of all. But every
set, even the largest one, has a powerset which contains it, and that
which contains it must be larger (a contradiction). The obvious solution:
define an extension of set theory incorporating two senses of
“containment” which work together in such a way that the largest set
can be defined as "containing" its powerset in one sense while
being contained by its powerset in the other. Thus, it
topologically includes itself in the act of descriptively including itself
in the act of topologically including itself..., and so on, in the course
of which it obviously becomes more than just a set. In the Cognitive-Theoretic
Model of the Universe or CTMU, the set of all sets, and the real
universe to which it corresponds, take the name (SCSPL) of the required extension of set theory. SCSPL, which
stands for Self-Configuring Self-Processing Language, is just a
totally intrinsic, i.e. completely self-contained, language that is
comprehensively and coherently (self-distributively) self-descriptive, and
can thus be model-theoretically identified as its own universe or
referent domain. Theory and object go by the same name because unlike
conventional ZF or NBG set theory, SCSPL hologically infuses sets and their elements with the distributed
(syntactic, metalogical) component of the theoretical framework containing
and governing them, namely SCSPL syntax itself, replacing ordinary
set-theoretic objects with SCSPL syntactic operators. The
CTMU is so-named because the SCSPL universe, like the set of all sets,
distributively embodies the logical syntax of its own descriptive
mathematical language. It is thus not only self-descriptive in
nature; where logic denotes the rules of cognition (reasoning,
inference), it is self-cognitive as well. (The terms "SCSPL"
and "hology" are explained further below; to skip immediately to
the explanations, just click on the above links.) An act is a temporal process, and
self-inclusion is a spatial relation. The act of self-inclusion is
thus "where time becomes space"; for the set of all sets, there
can be no more fundamental process. No matter what else happens in
the evolving universe, it must be temporally embedded in this dualistic
self-inclusion operation. In the CTMU, the self-inclusion process is
known as conspansion and occurs at the distributed, Lorentz-invariant
conspansion rate c, a time-space conversion factor already familiar
as the speed of light in vacuo (conspansion
consists of two alternative phases accounting for the wave and particle
properties of matter and affording a logical explanation for accelerating
cosmic expansion). When we imagine a dynamic self-including set, we
think of a set growing larger and larger in order to engulf itself from
without. But since there is no "without" relative to the real
universe, external growth or reference is not an option; there can be no
external set or external descriptor. Instead, self-inclusion and
self-description must occur inwardly as the universe stratifies into a
temporal sequence of states, each state topologically and computationally
contained in the one preceding it (where the conventionally limited term computation
is understood to refer to a more powerful SCSPL-based concept, protocomputation,
involving spatiotemporal parallelism). On the present level of discourse,
this inward self-inclusion is the conspansive basis of what we call spacetime. Every object in spacetime includes the
entirety of spacetime as a state-transition syntax according to
which its next state is created. This guarantees the mutual consistency of
states and the overall unity of the dynamic entity the real universe.
And because the sole real interpretation of the set-theoretic entity
"the set of all sets" is the entire real universe, the
associated foundational paradoxes are resolved in kind (by attributing
mathematical structure like that of the universe to the pure,
uninterpreted set-theoretic version of the set of all sets). Concisely,
resolving the set-of-all-sets paradox requires that (1) an endomorphism or
self-similarity mapping D:S-->rÎS
be defined for the set of all sets S and its internal points r; (2) there
exist two complementary senses of inclusion, one topological [S Ét
D(S)] and one predicative [D(S)
Éd
S], that allow the set to
descriptively "include itself" from within, i.e. from a state of
topological self-inclusion (where Ét
denotes topological or set-theoretic inclusion and Éd
denotes
descriptive inclusion, e.g. the inclusion in a language of its referents);
and (3) the input S of D be global and structural, while the output D(S) =
(r Éd
S)
be internal to S and play a syntactic role. In short, the set-theoretic
and cosmological embodiments of the self-inclusion paradox are resolved by
properly relating the self-inclusive object to the descriptive syntax in
terms of which it is necessarily expressed, thus effecting true
self-containment: "the universe (set of all sets) is that which topologically
contains that which descriptively contains the universe (set of all
sets)." This characterizes a system that consistently
perceives itself and develops its own structure from within via hology,
a 2-stage form of self-similarity roughly analogous to holography. (Hology
is a logico-cybernetic form of self-similarity in which the global
structure of a self-contained, self-interactive system doubles as its
distributed self-transductive syntax; it is justified by the obvious fact
that in a self-contained system, no other structure is available for that
purpose.) The associated conspansive
mapping D is called incoversion in the spatiotemporally inward
direction and coinversion in the reverse (outward, D-1)
direction. Incoversion carries global structure inward as
state-recognition and state-transformation syntax, while coinversion
projects syntactic structure outward in such a way as to recognize
existing structure and determine future states in conformance with it.
Incoversion is associated with an operation called requantization,
while coinversion is associated with a complementary operation called inner
expansion. The alternation of these operations, often referred to as wave-particle
duality, comprises the conspansion process. The Principle of
Conspansive Duality then says that what appears as cosmic expansion
from an interior (local) viewpoint appears as material and temporal
contraction from a global viewpoint. Because metric
concepts like "size" and "duration" are undefined
with respect to the universe as a whole, the spacetime metric is defined
strictly intrinsically, and the usual limit of cosmological regress, a
pointlike cosmic singularity, becomes the closed spacetime algebra already
identified as SCSPL. Thus, the real universe is not a static set,
but a dynamic process resolving the self-inclusion paradox. Equivalently,
because any real explanation of reality is contained in reality itself,
reality gives rise to a paradox unless regarded as an inclusory
self-mapping. This is why, for example, category theory is increasingly
preferred to set theory as a means of addressing the foundations of
mathematics; it centers on invariant relations or mappings between
covariant or contravariant (dually related) objects rather than on static
objects themselves. For similar reasons, a focus on the relative
invariants of semantic processes is also well-suited to the formulation of
evolving theories in which the definitions of objects and sets are subject
to change; thus, we can speak of time and space as equivalent to cognition
and information with respect to the invariant semantic relation processes,
as in "time processes space" and "cognition processes
information". But when we define reality as a process, we must
reformulate containment accordingly. Concisely, reality theory becomes a
study of SCSPL autology naturally formulated in terms of mappings. This is
done by adjoining to logic certain metalogical principles, formulated in
terms of mappings, that enable reality to be described as an autological
(self-descriptive, self-recognizing/self-processing) system. The first such principle is MAP,
acronymic for Metaphysical Autology Principle. Let S be the real
universe, and let T = T(S) be its theoretical description or
"TOE". MAP, designed to endow T and S with mathematical closure,
simply states that T and S are closed with respect to all internally
relevant operations, including recognition and description. In terms of
mappings, this means that all inclusional or descriptive mappings of S are
automorphisms (e.g., permutations or foldings) or endomorphisms
(self-injections). MAP is implied by the unlimited scope, up to perceptual
relevance, of the universal quantifier implicitly attached to reality by
the containment principle. With
closure thereby established, we can apply techniques of logical reduction
to S without worrying about whether the lack of some external necessity
will spoil the reduction. In effect, MAP makes T(S) "exclusive
enough" to describe S by excluding as a descriptor of S anything not
in S. But there still remains the necessity of providing S with a
mechanism of self-description. This mechanism is provided by another
metalogical principle, the M=R or Mind Equals Reality
Principle, that identifies S with the extended cognitive syntax D(S) of
the theorist. This syntax (system of cognitive rules) not only determines
the theorist's perception of the universe, but bounds his cognitive
processes and is ultimately the limit of his theorization (this
relates to the observation that all we can directly know of reality are
our perceptions of it). The reasoning is simple; S determines the
composition and behavior of objects (or subsystems) s in S, and thus
comprises the general syntax (structural and functional rules of S) of
which s obeys a specific restriction. Thus, where s is an ideal
observer/theorist in S, S is the syntax of its own observation and
explanation by s. This is directly analogous to "the real universe
contains all and only that which is real", but differently stated:
"S contains all and only objects s whose extended syntax is
isomorphic to S." M=R identifies S with the veridical limit of any
partial theory T of S [limT(S) = D(S)], thus making S "inclusive
enough" to describe itself. That is, nothing relevant to S is
excluded from S @
D(S). Mathematically, the M=R Principle is
expressed as follows. The universe obviously has a structure S. According
to the logic outlined above, this structure is self-similar; S distributes
over S, where "distributes over S" means "exists without
constraint on location or scale within S". In other words, the
universe is a perfectly self-similar system whose overall structure is
replicated everywhere within it as a general state-recognition and
state-transition syntax (as understood in an extended computational
sense). The self-distribution of S, called hology,
follows from the containment principle,
i.e. the tautological fact that everything within the real universe must
be described by the predicate "real" and thus fall within the
constraints of global structure. That this structure is completely
self-distributed implies that it is locally indistinguishable for
subsystems s; it could only be discerned against its absence, and it is
nowhere absent in S. Spacetime is thus transparent from within, its
syntactic structure invisible to its contents on the classical
(macroscopic) level. Localized systems generally express and utilize only
a part of this syntax on any given scale, as determined by their specific
structures. I.e., where there exists a hological incoversion endomorphism
D:Sà{rÎS}
carrying the whole structure of S into every internal point and region of
S, objects (quantum-geometrodynamically) embedded in S take their
recognition and state-transformation syntaxes directly from the ambient
spatiotemporal background up to isomorphism. Objects thus utilize only
those aspects of D(S) of which they are structural and functional
representations. The inverse D-1 of this map (coinversion)
describes how an arbitrary local system s within S recognizes S at the
object level and obeys the appropriate "laws", ultimately giving
rise to human perception. This reflects the fact that S is a
self-perceptual system, with various levels of self-perception emerging
within interactive subsystems s (where perception is just a refined form
of interaction based on recognition in an extended
computational sense). Thus, with respect to any class {s} of subsystems of
S, we can define a homomorphic submap d of the endomorphism D: d:Sà{s} expressing only that
part of D to which {s} is isomorphic. In general, the si are
coherent or physically self-interactive systems exhibiting dynamical and
informational closure; they have sometimes-inaccessible internal
structures and dynamics (particularly on the quantum scale), and are
distinguishable from each other by means of informational boundaries
contained in syntax and comprising a "spacetime metric". According to the above definitions, the
global self-perceptor S is amenable to a theological interpretation, and
its contents {s} to "generalized cognitors" including subatomic
particles, sentient organisms, and every material system in between.
Unfortunately, above the object level, the validity of s-cognition
- the internal processing of sentient subsystems s - depends on the
specific cognitive functionability of a given s...the extent to which s
can implicitly represent higher-order relations of S. In General
Relativity, S is regarded as given and complete; the laws of mathematics
and science are taken as pre-existing. On the quantum scale, on the other
hand, laws governing the states and distributions of matter and energy do
not always have sufficient powers of restriction to fully determine
quantum behavior, requiring probabilistic augmentation in the course of
quantum wavefunction collapse. This prevents a given s, indeed anything
other than S, from enclosing a complete nomology (set of laws); while a
complete set of laws would amount to a complete deterministic history of
the universe, calling the universe "completely deterministic"
amounts to asserting the existence of prior determinative constraints. But
this is a logical absurdity, since if these constraints were real, they
would be included in reality rather than prior or external to it (by the containment
principle). It follows that the universe freely determines its own
constraints, the establishment of nomology and the creation of its
physical (observable) content being effectively simultaneous and
recursive. The incoversive distribution of this relationship is the basis
of free will, by virtue of which the universe is freely created by
sentient agents existing within it. Let's elaborate a bit. Consider the universe
as a completely evolved perceptual system, including all of the
perceptions that will ultimately comprise it. We cannot know all of those
perceptions specifically, but to the extent that they are interactively
connected, we can refer to them en masse. The set of
"laws" obeyed by the universe is just a minimal set of logical
relations that suffices to make these perceptions noncontradictory, i.e.
mutually consistent, and a distributed set of laws is just a set of laws
formulated in such a way that the formulation can be read by any part of
the system S. Obviously, for perceptions to be connected by laws, the laws
themselves must be internally connected according to a syntax, and the
ultimate syntax of nomological connectedness must be globally valid;
whatever the laws may be at any stage of system evolution, all parts of S
must be able to unambiguously read them, execute and be acted upon by
them, and recognize and be recognized as their referents
("unambiguously" implies that 2-valued logic is a primary
ingredient of nomology; its involvement is described by a third
metalogical principle designed to ensure consistency, namely MU or Multiplex
Unity). This implies that the action and content of the laws are
merged together in each part of the system as a single (but dual-aspect)
quantity, infocognition. The connectedness and consistency of
infocognition is maintained by refinement and homogenization as
nomological languages are superseded by extensional metalanguages in order
to create and/or explain new data; because the "theory" SCSPL
model-theoretically equates itself to the real universe, its
"creation" and causal "explanation" operations are to
a certain extent identical, and the SCSPL universe can be considered to
create or configure itself by means of "self-theorization" or
"self-explanation". The simplest way to explain
"connected" in this context is that every part of the
(object-level) system relates to other parts within an overall structural
description of the system itself (to interpret "parts", think of
events rather than objects; objects are in a sense defined on
events in a spatiotemporal setting). Obviously, any part which fails to
meet this criterion does not conform to a description of the system and
thus is not included in it, i.e. not "connected to" the system
(on the other hand, if we were to insist that it is included or
connected, then we would have to modify the systemic description
accordingly). For this description to be utile, it should be maximally
compact, employing compact predictive generalizations in a regular way
appropriate to structural categories (e.g., employing general "laws
of physics"). Because such laws, when formulated in an "if
conditions (a,b,c…) exist, then (X and Y or Z) applies" way, encode
the structure of the entire system and are universally applicable within
it, the system is "self-distributed". In other words, every part
of the system can consistently interact with every other part while
maintaining an integral identity according to this ("TOE")
formulation. Spatiotemporal relations can be skeletally depicted as edges
in a graph whose vertices are events (physical interactions), i.e.
spacetime "points". In this sense, graph-theoretic connectivity
applies. But all those object-level connections must themselves be
connected by more basic connections, the basic connections must be
connected by even more basic connections, and so on. Eventually - perhaps
sooner than later - we reach a basic level of connectivity whose syntax
comprises a (partially undecidable) "ultimate nomology" for the
level of reality we’re discussing. Is this nomology, and the cognitive syntax in
which it is expressed, wholly embodied by matter? In one sense the
answer is yes, because S is distributed over each and every material sÎS
as the reality-syntax D(S). Thus, every axiom and theorem of mathematics
can be considered implicit in material syntax and potentially exemplified
by an appropriate material pattern, e.g. a firing of cerebral neurons.
Against holism - the idea that the universe is more than the sum of its
parts - one can further object that the holistic entity in question is
still a material ensemble, thus insinuating that even if the universe is
not the "sum" of its parts, it is still a determinate function
of its parts. However, this fails to explain the mutual consistency of
object-syntaxes, without the enforcement of which reality would
disintegrate due to perceptual inconsistency. This enforcement function
takes matter as its argument and must therefore be reposed in spacetime
itself, the universal substrate in which matter is unconditionally
embedded (and as a geometrodynamic or quantum-mechanical excitation of
which matter is explained). So the background has logical ascendancy over
derivative matter, and this permits it to have aspects, like the power to
enforce consistency, not expressible by localized interactions of compact
material objects (i.e., within the bounds of materialism as invoked
regarding a putative lack of "material evidence" for God,
excluding the entire material universe). On the other hand, might cognitive syntax
reside in an external "ideal" realm analogous to Plato's world
of Parmenidean forms? Plato’s ideal abstract reality is explicitly set
apart from actual concrete reality, the former being an eternal world of
pure form and light, and the latter consisting of a cave on whose dirty
walls shift murky, contaminated shadows of the ideal world above. However,
if they are both separate and in mutual correspondence, these two
realities both occupy a more basic joint reality enforcing the
correspondence and providing the metric of separation. If this more basic
reality is then juxtaposed to another, then there must be a more basic
reality still, and so on until finally we reach the most basic level of
all. At this level, there will (by definition) be no separation between
the abstract and concrete phases, because there will be no more basic
reality to provide it or enforce a remote correspondence across it. This
is the inevitable logical terminus of "Plato’s regress". But
it is also the reality specified by the containment
principle, the scope of whose universal quantifier is unlimited up to
perceptual relevance! Since it is absurd to adopt a hypothesis whose
natural logical extension is a negation of that hypothesis, we must assume
that the ideal plane coincides with this one…but again, not in a way
necessarily accessible to identifiable physical operations. Rather,
physical reality is embedded in a more general or "abstract"
ideal reality equating to the reality-syntax D(S), and the syntax D(S) is
in turn embedded in physical reality by incoversion. Thus, if D(S)
contains supraphysical components, they are embedded in S right along with
their physical counterparts (indeed, this convention is already in
restricted use in string theory and M-theory, where unseen higher
dimensions get "rolled up" to sub-Planck diameter). What does this say about God? First,
if God is real, then God inheres in the comprehensive reality syntax, and
this syntax inheres in matter. Ergo, God inheres in matter, and indeed in
its spacetime substrate as defined on material and supramaterial levels.
This amounts to pantheism, the thesis that God is omnipresent with
respect to the material universe. Now, if the universe were pluralistic
or reducible to its parts, this would make God, Who coincides with the
universe itself, a pluralistic entity with no internal cohesion. But
because the mutual syntactic consistency of parts is enforced by a unitary
holistic manifold with logical ascendancy over the parts themselves -
because the universe is a dual-aspected monic entity consisting of
essentially homogeneous, self-consistent infocognition - God retains
monotheistic unity despite being distributed over reality at large. Thus,
we have a new kind of theology that might be called monopantheism,
or even more descriptively, holopantheism. Second, God is indeed
real, for a coherent entity identified with a self-perceptual universe is
self-perceptual in nature, and this endows it with various levels of
self-awareness and sentience, or constructive, creative
intelligence. Indeed, without a guiding Entity whose Self-awareness
equates to the coherence of self-perceptual spacetime, a self-perceptual
universe could not coherently self-configure. Holopantheism is the
logical, metatheological umbrella beneath which the great religions of
mankind are unknowingly situated. Why, if there exists a spiritual metalanguage
in which to establish the brotherhood of man through the unity of
sentience, are men perpetually at each others' throats? Unfortunately,
most human brains, which comprise a particular highly-evolved subset of
the set of all reality-subsystems, do not fire in strict S-isomorphism
much above the object level. Where we define one aspect of
"intelligence" as the amount of global structure functionally
represented by a given sÎS,
brains of low intelligence are generally out of accord with the global
syntax D(S). This limits their capacity to form true representations of S
(global reality) by syntactic autology [d(S) Éd
d(S)] and make rational ethical calculations. In this sense, the vast
majority of men are not well-enough equipped, conceptually speaking, to
form perfectly rational worldviews and societies; they are deficient in
education and intellect, albeit remediably so in most cases. This is why
force has ruled in the world of man…why might has always made right,
despite its marked tendency to violate the optimization of global utility
derived by summing over the sentient agents of S with respect to space and
time. Now, in the course of employing deadly force
to rule their fellows, the very worst element of humanity – the
butchers, the violators, i.e. those of whom some modern leaders and
politicians are merely slightly-chastened copies – began to consider
ways of maintaining power. They lit on religion, an authoritarian
priesthood of which can be used to set the minds and actions of a populace
for or against any given aspect of the political status quo. Others,
jealous of the power thereby consolidated, began to use religion to gather
their own "sheep", promising special entitlements to those who
would join them…mutually conflicting promises now setting the promisees
at each other’s throats. But although religion has often been employed
for evil by cynics appreciative of its power, several things bear notice.
(1) The abuse of religion, and the God concept, has always been driven by
human politics, and no one is justified in blaming the God concept,
whether or not they hold it to be real, for the abuses committed by evil
men in its name. Abusus non tollit usum. (2) A religion must
provide at least emotional utility for its believers, and any religion
that stands the test of time has obviously been doing so. (3) A credible
religion must contain elements of truth and undecidability, but no
elements that are verifiably false (for that could be used to overthrow
the religion and its sponsors). So by design, religious beliefs generally
cannot be refuted by rational or empirical means. Does the reverse apply? Can a denial of
God be refuted by rational or empirical means? The short answer is yes;
the refutation follows the reasoning outlined above. That is, the above
reasoning constitutes not just a logical framework for reality theory, but
the outline of a logical proof of God's existence and the basis of a
"logical theology". While the framework serves other useful
purposes as well, e.g. the analysis of mind and consciousness, we'll save
those for another time.
© 1998 by Christopher Michael Langan (All Rights Reserved) |
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