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INVERTED THEORY NETWORKS by Nico Christodoulides, 2005 Nico@Christodoulides.net |
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Abstract |
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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 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. |