http://www.christodoulides.net/


And here follows the blurb:

INVERTED THEORY NETWORKS
© Copyright by Nico Christodoulides, 2005

Perhaps some of you might find this interesting. Proudly South African and good to see some interesting work stemming from my old uni.

Abstract

The logicatom is defined, and it is argued that this represents the
quantum of knowledge. Theory networks encapsulating a set of
logicatoms and the dynamic relations between them, are defined. It is
shown that these structures can emulate cellular automaton systems and
in particular, simulate universal Turing machines. The regulating
principle of natural selection is formalised together with its
necessary and sufficient conditions. This is achieved using the
equation of natural selection. It is proven that there exist inverted
theory networks (an analogous construct to theory networks) that
satisfy all the requirements specified for natural selection to
regulate their dynamics. The applicability of inverted theory networks
to modelling thought is analysed. Further, inverted theory networks
are proposed as a candidate for the pregeometry hypothesised by
Wheeler.

Overview

Bertrand Russell, Richard Dawkins and John Archibald Wheeler provide
me with the postulates from whence this thesis arises.

Russell's Logical Atomism

The primary hypothesis adopted in this thesis is that of logical
atomism. Ludwig Wittgenstein and Bertrand Russell were the prime
exponents of this philosophy. The logical atomism view of reality
assumes that all knowledge must begin with sensory experience. Genuine
information about the world must be acquired by a posteriori means, so
that nothing can be thought without first being sensed. From this
beginning, Russell argued that everything else follows by logical
analysis. Simple facts like 'It is raining' are the atomic facts or
'logical atoms' upon which all human knowledge is grounded. In
particular, Russell claims in the fifth chapter of The Problems of
Philosophy (1912): "Every proposition which we can understand must be
composed wholly of constituents with which we are acquainted." This
statement forms the founding argument in the formal definition of a
'logical atom'.

Dawkins' Meme

Richard Dawkins hypothesised the existence of a 'meme'. In order to
define a meme, he referred to the analogous construct, the gene.
Biological organisms are defined by their genotype and their
phenotype. The genotype (nucleic acids) represents the underlying
genetic coding while the phenotype (proteins) is the expression of the
genotype within an environment. Dawkins defined the meme as "a unit of
information residing in the human brain". Just as the phenotype of a
particular gene complex in a species determines a particular trait
e.g. blue eyes in human beings, the phenotype of a meme complex
represents a concept that can be understood, learnt or sensed. This
can be represented as a collection of words, music or visual images.
One can view Dawkins meme as equivalent to Russell's logical atom.
However, Dawkins' genius came in observing the regulating principle of
these entities. Dawkins hypothesised that the dynamic behaviour of
memes is governed by natural selection. From an intuitive perspective,
consider the following example. This thesis represents the phenotype
of a meme complex existing in the author's brain. By reading it, the
reader has allowed the meme complex to make a copy of itself in the
reader's brain. Thus memes have the property of reproduction. Now the
reader will understand this thesis in a different way to the author
(or any other reader for that matter) due to the incoming knowledge
interacting with the existing knowledge in the reader's head. Thus the
meme complex can be said to have mutated as a copy was made. Finally,
depending on whether the reader thinks this thesis is of any value to
the scientific community or not, he/she may recommend others to read
it, or he/she might forget entirely about it. Thus the meme complex
exhibits the property of differential fitness i.e. its spread and
survival depends upon its makeup - in this particular instance, its
acceptance within the scientific community. This fitness may be
quantified by the number of citations in future scientific work. The
three properties stated in bold are exactly the necessary requirements
of natural selection The second hypothesis is encapsulated in the
statement: 'Natural selection acts on memes and regulates their
survival, resulting in the fittest meme surviving.'

Wheeler's Pregeometry

Einstein's theory of general relativity elevated the importance of the
underlying spacetime structure in physics. Prior to the theory, the
spacetime continuum was regarded as the arena in which the laws of
physics act. Einstein's field equations dictated that energy curved
spacetime and spacetime in turn prescribed the dynamics of classical
energy. In Wheeler's words, general relativity "dethroned spacetime
from a post of preordained perfection high above the battles of matter
and energy, and marked it as a new dynamic entity participating
actively in this combat." What was previously perceived as a
gravitational force field is now known to be the effects of curved
spacetime. Further, physical laws such as the conservation of energy
and momentum ended up being a mathematical consequence of the
geometry. Misner and Wheeler took these beautiful concepts to the next
logical step by asking the question: 'Is the spacetime continuum all
there is to physics?' In other words, can curved spacetime solely
represent all the laws of physics. To answer this question, the theory
of geometrodynamics was born. Geometrodynamics is the study of the
geometry of curved empty space and the relative dynamics of subspaces
therein, as prescribed by the Einstein field equations. Misner and
Wheeler went some way to show that classical physics embodying
gravitation, electromagnetism, non-quantised charge and non-quantised
mass can be represented as purely geometrical phenomena. This theory
reached its explanatory limit when attempting to discuss quantum
phenomena. The limitation in the theory was identified in that it was
constrained to operate in a differentiable manifold. There was no
natural way of modelling the dynamics in the underlying topology. To
overcome this barrier, Wheeler hypothesised the existence of a
'pregeometry'. Wheeler argued that spacetime itself must be understood
in terms of the more fundamental structure. The underlying principle
of such a structure was to be in its simplicity. In particular,
Wheeler stated: "All of physics, in my view, will be seen someday to
follow the pattern of thermodynamics and statistical mechanics, of
regularity based on chaos, of 'law without law'. Specifically, I
believe that everything is built higgledy-piggledy on the
unpredictable outcomes of billions and billions of elementary quantum
phenomena, and that the laws and initial conditions of physics arise
out of this chaos by the action of a regulating principle, the
discovery and proper formulation of which is the number one task of
the coming third era of physics."

Problem statement and thesis objective

These disparate topics are linked in the following way: Russell's
logical atom is analogous to Dawkins' meme. Russell's postulate
describes properties regarding the quanta of thought; Dawkins
postulates what regulates these quanta. The question as to what this
has to do with physics and Wheeler's pregeometry, comprising the
hypothesised fundamental building blocks of physical law was elegantly
answered by G.F.R. Ellis: "Human thoughts can cause real physical
effects." If I have the intention of picking up a stone and throwing
it, the result would be the physical effect of a stone hurtling
through the air. "At present there is no way to express this
interaction in the language of physics, even though our causal schemes
are manifestly incomplete if this is not taken into account. The
minimum requirement to do so is to include the relevant variables in
the space of variables considered. That then makes these variables and
their effects a part of physics - or perhaps of fundamental physics".
Thus Wheeler's pregeometry must comprise the 'variables' that model
intent i.e. thought. My hypothesis is that the structure that models
the quanta of thought is Wheeler's hypothesised pregeometry. Further,
the regulating principle sought after by Wheeler is none other than
Darwin's law of natural selection, originally suggested in 1859 as the
principle mechanism of evolutionary change.

The above paragraph guided my research program resulting in the
question "Can a formal paradigm be created in which to model these
postulates?". I split this problem statement up into 2 objectives: The
construction objective encompasses defining a formal mathematical
space comprised of entities that represent Dawkins' memes or Russell's
logical atoms. Further, this objective encapsulates showing that the
dynamics of the space is regulated by the principle of natural
selection. The application analysis objective encompasses
investigating if this space can be applied to analysing the dynamical
properties of knowledge and whether it serves as a candidate for
modelling pregeometries in physics. Needing to define a space that
comprises a set of elements representing knowledge, I naturally enter
the formal arena of knowledge representation – description logics.
Research within this broad mathematical arena is guided by Russell's
claim that "every proposition which we can understand must be composed
wholly of constituents with which we are acquainted". This is
interpreted as saying that 'new' knowledge is made up of 'existing'
knowledge i.e. all knowledge is comprised of knowledge. I use this to
define the basic entity of my space – the logicatom. I then proceed to
formally construct platforms comprising dynamic sets of these entities
i.e. theory networks and inverted theory networks. In order to prove
that these structures are regulated by natural selection, I derive the
necessary and sufficient requirements for it to be said that natural
selection regulates the dynamics of a space. The construction
objective is met through the construction of a particular inverted
theory network, whereupon I prove that it is regulated by natural
selection. The application analysis objective is met since it
completely guided the construction of the space under consideration.
Various case studies are given throughout the thesis that show the
various applications of these structures to multiple fields of study.