“Reality leaves a lot to the imagination.” John Lennon
In this section I will discuss the ideas and concepts concerning ‘reality’ with the intention of developing a model of realities that is pertinent to the development of a computational model of the universe. As in all of the research a ‘pragmatic ‘approach is taken to the understanding of reality and hence this is by no means a comprehensive treatise on the subject.
A View of Realities
I think that there is no better place to start discussing ‘reality’ than through the work of Rene Descartes, who is by many considered to be the father of the philosophy of science. Descartes identified two categories or types of reality that I will term internal and external realities. Internal reality is evident from the Cartesian proposition; I think therefore I exist’. This ‘truth’ was identified by Rene Descartes who used what is known as Cartesian or hyperbolic doubt to identify statements that were beyond doubt. From his famous statement it follows that internal reality depends upon conscious existence. The ontology of internal reality includes sensory perceptions and ideas generated by the conscious mind. Therefore I can include sounds, smells, emotional states and even the ideas in this essay as part of my internal reality. Part of the ontology of my internal reality is what I would call my internal physical reality; by this I mean the universe that I perceive through my senses and that I interact with on a moment-by-moment basis. This precludes aspects of the universe that I may perceive through my imagination or dreams. This physical reality that I experience is a combination of my self-awareness, my perception through my senses and my thought processes. All of these aspects enable me to be conscious of an internal physical reality in which I exist. This physical reality is an internal reality to me. When I have no consciousness this internal physical reality no longer exists. My internal physical reality is only part of my entire consciousness as this also includes other aspects of which I am aware, such as emotional feelings and dreams. This internal physical reality includes the perception of time and as such it only consists of what has been and what is. So my internal physical reality does not include potential events that I may or may not perceive in the future as these would be dreams or created through imagination. It is worth noting at this point that I talk of my consciousness and internal reality because if I apply hyperbolic doubt then I can doubt that other beings have consciousness because I perceive them through my unreliable senses.
Descartes also supported the proposition of an external reality. He argued that sensory perceptions came to him involuntarily and were therefore external to his senses and therefore this was evidence of something existing outside of the conscious mind. To bastardise a quotation by Philip K Dick, “external reality is that which, when you stop perceiving it, it doesn’t go away.” There are philosophical arguments that raise doubt regarding this proposition; one of the strongest is the movement known as ‘scepticism’. Now most physicists would acknowledge the concept of an external reality as would I, in fact it is the objective of the theory to elaborate on what the structure of such a reality may be, but I have to acknowledge that this is not a statement of truth but merely a proposition. It is also worth noting at this point that although Descartes supported the idea of an external reality he argued that because he doubted the source of his experiences he therefore doubted anything derived from his senses and hence he could not know anything about the external reality.
I now want to discuss the relationship between the internal conscious reality and the external ‘physical’ reality. Unfortunately, if hyperbolic doubt is applied to this question then there seem to be no answers that are beyond doubt. However in the hope of making some progress I would postulate that the ontology of the external reality includes aspects of me and that even when I cease to exist, those parts of the ontology of the external reality continue to exist albeit in a different form. This seems a reasonable viewpoint that is supported by the fact that physics indicates that I can perceived myself as being made of the same ‘stuff’ as other perceived objects in the universe that themselves would be part of the ontology of the external reality. The biggest debate in this area is that of consciousness itself. Descartes supported a dualist viewpoint in that he considered his body to be a machine that had extension and motion and that followed the perceived laws of physics whereas he considered the mind to be non-material and lacking extension and motion and not following the laws of physics. This viewpoint suggests that the mind is not part of the ontology of the external reality and it may well be considered to have a reality of its own. Later I will discuss the importance of aspects of the science of complex systems and this can lead to a view that the mind is an emergent property of our complex and physical brains. Such a position would suggest that the mind could be part of the ontology of the external reality. The answer to this question is thankfully irrelevant to most of the research to be discussed; however the important conclusion to be made is that we seem to perceive the external reality from within that reality using our senses as input. Some meditative philosophies would even argue against this view; however such arguments will not be pursued here.
The view of realities that I have given is similar to a philosophical approach termed ‘causal realism’. Causal realism assumes there to be an external reality and that we acquire beliefs through our external being interacting with other aspects of that reality and this leads to an acquisition of information that is processed by the brain to create a belief. In this approach the route by which I acquire beliefs is important. For example for me to ‘see’ an object it is essential that the object is the cause of the beliefs I acquire about it. That set of beliefs is my internal reality.
This approach is generally considered to be the most sustainable viewpoint. However this short foray into reality shows that once one goes beyond Descartes’ premise one has to make assumptions and therefore the possibility of a singular truth regarding realities disappears.
So to recap the key points regarding realities in relation to the theory that will be developed are as follows:
There is an internal reality that is unique to me and exists whilst I have conscious existence. This is the starting point for such discussion and is as such without doubt.
There is an external reality that exists beyond my conscious existence. This proposition can be doubted but it is ‘believed’ to be true for the purposes of this research.
I am in some way part of the ontology of the external reality but with unresolved questions regarding the inclusion of aspects of my consciousness within that ontology.
Part of the ontology of my internal reality is my internal physical reality that is my perception of the interactions with the external reality through my senses.
I perceive the external reality through my senses (my internal physical reality) and from within the external reality.
The Importance of Structure and Pattern
One of the only common characteristic that can be inferred between the internal and external realities is that they are likely to both have structure in the broadest sense of its meaning. In this context ‘pattern’ is a conscious representation of structure. To everything in my internal reality I can associate a pattern in some manner whether it is tangible or not, for example objects in my internal reality can have many perceived patterns some of which may be associated with mathematical constructs and others with constructs from physics, others may be just visual representations or the relationship between objects within my belief system. I would also postulate that the external reality has structure in some manner otherwise the most fundamental concepts of objects, relationships and interactions between those objects would make no sense. My reason for introducing the ideas of structure and pattern at this low level is that they will play an important part in the broad analysis of the results of the theory to be described.
A concept of structure can be applied to the internal and external reality.
Pattern is a conscious representation of a structure.
The Importance of Complexity
It is only in recent decades that we have understood how seemingly insignificant interactions between parts of a system can have a dramatic effect upon the system’s evolution and behaviour at various scales. Well into the 20th century scientist still attributed unexpected results in experiments to validate theories as ‘experimental error’. But it turned out that most of these observations were due to the extraordinarily complex behavior of even perceptually simple systems. In the last few decade a science has emerged that is trying to understand the nature of complexity and general characteristics of complex systems, whether the system is the climate, or the variations in populations of specific herds of animals.
We now know that the complexity of the behaviour and structure of a system limits what we can know about that system. It is also true that the behaviour changes with scale and so we perceive our universe at scales where things seem predictable and at which we can associate a mathematical description. Such scales are relatively large and are predictable through what we call classical physics. In general the science of complexity suggests that at smaller scales the level of predictability decreases and our internal physical reality becomes more uncertain and doubt increases, meaning that comprehension of an external reality moves further away. At these smaller scales we use modern physics such as quantum mechanics and we find unsurprisingly that they introduce inherent uncertainty and a seeming limit to our understanding.
It may be that at the quantum scale we perceive the interactions at which the nature and structure of the external reality is manifest and we may find that it is complexity that is the ultimate arbiter on what we can know about both our internal and external realities.
Logic and its Importance in Shaping our Thoughts
Now that I have developed a fundamental model of realities I want to look at how mathematics and physics view reality and then draw some conclusions about those viewpoints based upon the model that I have described.
For me there is a clear hierarchy of knowledge that both enables and limits my perception of internal reality. To this end I would assert that it is the knowledge of logic that underpins mathematics and the knowledge of mathematics that underpins physics. So in the next sections I want to look at each of these areas of knowledge and discuss the relationships between them and what they have to say about reality as I have described it. You may notice that I start to use ’we’ in this section as I am willing to forgo my doubt of your existence for the purposes of a more readable text.
Logic is usually defined as the science of reasoning or correct inference and is a method of reasoning that attempts to identify statements that are either true or false. In this context ‘true’ means a statement that is consistent within the rules of the logic being used. This means that even using our most analytic methods of reasoning the idea of ‘truth’ is always a relative concept and therefore statements may be true in one logic method but be un-indefinable or false in another.
There is a second chink in the armour of any logical system; from any logical system we can identify a paradox, a paradox that would cause any computer trying to resolve it to go into melt down. This paradox can be presented in a formal and generalised manner known as Russell’s Paradox (need a reference). The popular way of presenting this paradox is the story of the Barber of Seville: “A man of Seville is shaved by the Barber of Seville if and only if the man does not shave himself. Does the Barber shave himself?” If he does then he doesn’t, but if he doesn’t then he does! Interestingly in the interactions that we have with the external reality we never see such paradoxes, for to do so would require a physical object to be or contain itself!
A question arises at this point as to whether this is telling us something fundamental about the relationship between logic that we are cognoscente of and the external reality? It would seem to me the lack of perceived paradoxes in the internal physical reality indicates that our formulations of logic are in some ways limited in comparison to those that may underpin an external reality. It is possible that there are aspects of an external reality that we are not aware of, such as extra dimensions, such extensions may enable the paradoxes in the underpinning logic to occur in a part of the external reality to which we are not privileged.
Logic systems still remain our best method of removing as much subjectivity from reasoning as possible, but the subjectivity still remains in the fundamental rules upon which the logic is based.
I believe logic to be the most important building block of scientific knowledge as mathematics and consequently physics accord with it, and to a significant extent they are derived from it. The other, and in some ways more important aspect of logic is that it has over many centuries shaped the way in which we perceive external reality and through mathematics the way in which we codify that reality.
Like all other knowledge, logic is part of our internal reality but how does it relate to the external reality. Well the fact that we perceive the external reality to be ‘logical’ in some manner seems to support the proposition that there is some logical foundation to external reality, however taking a Cartesian standpoint, we cannot know that logic, only that our internal representation is a full or partial representation of it. Whether it is a full or partial representation will be discussed in the context of physics and reality.
The important propositions are:
Logic is a formulation that exists in the internal reality.
Truth is always relative to the logic from which the assertion is being made.
All logic formulations have a fundamental paradox that cannot be resolved.
We select a logic formulation based upon our perception of the internal reality to which it will be applied.
Our perception of the external reality suggests that a form of logic exists in the external reality.
Mathematics and Reality
“Mathematics is imagination restricted to consistency”, David Valdman.
I now want to turn my attention to mathematics and its relationship to logic and to the realities that I have discussed in the previous sections. To do so I will concentrate on pure mathematics, that is, mathematics in its purest form, without the confusions of its application to physics as this will be discussed in the next section.
Mathematics can be viewed as a method of symbolic codification of patterns; by this I mean that you can consider any mathematical expression as representing a pattern and such patterns can be perceived at many levels of abstraction, for example the concept of ‘spherical’ can be codified and then altered to codify a specific spherical object. So mathematics can in some manner codify our physical internal reality at many levels of abstraction. Some of these codifications can in themselves generate other codifications and in essence be seen as pattern generators. Such codifications can become extremely abstract and represent patterns that we cannot relate to our perceived internal reality. Pure mathematics is concerned with all possible codifications, not only those that are of practical use in describing our internal reality.
Maths as Reality
Before embarking on a discussion about the foundations of mathematics one cannot disregard a powerful emotional quality that mathematics can elicit. For those who have done pure mathematics there can be a feeling of remoteness from perceptual understanding of the universe; strings of symbols can flow onto the paper generated purely from a world of abstract rules and methods that have no recourse to the universe around us. With this freedom from the infringement of worldly perceptions comes an emotional feeling that the mathematics has a life of its own. Penrose concedes that “to a mathematician both reality and mathematics itself has an emotional aspect. The process of doing maths can seem far removed from any real world relationship”. But however comforting this viewpoint may seem it is not a rational argument for the proposition that mathematics is a reality in itself.
The philosophical idea of mathematics being a reality in itself harks back to Plato and his idea of the world of mathematical forms and is the basis of the philosophy of mathematical realism. In this philosophy there is, for example a mathematical form that is a circle, but this is not the same as the physical representation of a circle that we perceive in our internal reality. In essence Plato argues that the external reality is partly or fully comprised of mathematical forms. In recent years Max Tegmark has taken Plato’s viewpoint further and proposed that external reality is a mathematical object. Tegmark’s argument is that the mathematics that we use to describe the universe through physics is becoming more extensive and precise whilst also becoming more abstract. Therefore if one is left with a complete theory of the universe that is abstractly mathematical then the universe IS a mathematical object.
I personally have issues with any proposition that external reality is some kind of mathematical object. To a mathematical object only has meaning if it is realised. This means that the mathematical object itself must be codified into our cerebral cortex, or in computer memory, or even just written on a piece of paper. Therefore such a mathematical object would need a substrate upon which it is realised and then we are back with the substrate being external reality. Also all mathematical objects that we have realised are static. They need the concepts of iterative processing through the substitution of variables by numbers to generate a set of results leading to a dynamic behaviour. Perhaps this viewpoint supports a computational model of external reality whereby mathematics and logic as we understand them would be instantiated into the fabric of external reality.
If mathematics is a reality in itself then it must be independent of our thought processes, in essence we discover mathematical forms as opposed to creating them. However there is no experiment yet devised that can answer this question. I believe that some of the characteristics of mathematics that I will discuss later indicate that mathematics has serious integral weaknesses. These weaknesses suggest to me that mathematics is not an independent reality but a human creative process of which some parts map onto our perception of the internal reality. This does not mean that mathematics cannot tell us something about an external reality as we will now go on to discover.
Mathematics and Logic
It can be difficult to identify where pure logic ends and pure mathematics begins as they share the common characteristics of being based upon a set of axioms, being capable of representing objects or patterns, and being formal methods of reasoning that are not constrained by our perceptions of the universe from within.
It is surprising, given this relationship between the two subjects that, with the exception of geometry, much of the development of mathematics prior to 1900 occurred without the application of formal logic systems. Geometry is the exception to this fact as it was rigorously constructed by Euclid using the logical rigor laid down my Aristotle. However arithmetic and other areas of mathematics tended to develop with little recourse to formal logical analysis. Part of the reason for this fracture between the subjects is down to human nature, unfortunately mathematics and logic fractured into two separate institutionalise subjects the members of whom focused on their own subjects.
In the early part of the twentieth century the formidable mathematician David Hilbert attempted to establish a secure logical foundation for mathematics by showing that there existed a complete and consistent set of axioms from which all mathematics could be derived. This endeavour led to the development of meta-mathematics, the study of mathematics using mathematical methods, which harks back to the ability of mathematics to codify itself. This act enabled logisticians to get involve in mathematics and mathematicians to use logic to analyse the foundations of their subject.
Through this exercise a logic was developed, called predicate logic that encompassed the assertions made in mathematics and computing, especially equivalence, such that A and B is equivalent to B and A.
It should be noted that the selection of Predicate Calculus is a purely practical one based upon the fact that it accords with our conception of internal physical reality.
For me, the most impressive example of the limitations of predicate calculus is that the fundamental principle of prediction (either, or and non-contradiction) is bivalent; a proposition is either true or false, black or white, here or there and it cannot be a bit true and a bit false, grey or here and there. This bivalent nature of predicate calculus fits the way in which we perceive our universe on a moment by moment basis. However around 1910 the development of quantum mechanics started to show that at the quantum level perceived reality is not true or false, black or white, here or there but instead can be true and false, black and white and here and there at the same time! This serves to prove that our use of suitable mathematics to explore internal reality is shifting sands. It is worth noting that to resolve the bivalent problem in quantum mechanics the application of a further abstraction in the form of category theory has led to the development of various topoi, whereby new consistent sets of axioms are identified, some of which are not bivalent; and there are suggestions that such a topoi may be a more suitable basis for the mathematics and logic required to describe quantum mechanical behaviour.
Probably the most important outcome of the development of predicate calculus was its use by Gödel in conjunction with the idea of Russell’s Paradox to show that most formulations of mathematics that one would consider useful are incomplete. Gödel in fact proposed two theorems; the second theorem states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency. This means that one can always find a paradox or inconsistency in the mathematics. One can add another axiom to the system to remove the problem but this just leads to another inconsistency. This theory has been interpreted to show that Hilbert’s idea of a secure basis for mathematics in logic cannot be completed. I shall return to this issue when discussing the problems with physics.
I believe that incompleteness may be identifying the fact that any mathematically described system, can only be described fully from outside of that system. That is, axioms are required to define the system that cannot be identified or even envisaged from within the system. That additional axiom(s) required for completeness is like stepping outside of the system being described and as we cannot step outside our universe our mathematics will be incomplete.
What this all seems to say is that in certain circumstances pure mathematics can be fundamentally flawed, and for me one of these areas is when discussing its relationship to any realities. It seems reasonable to us to consider that in some way an external reality will have mathematical characteristics and it is possible that due to Gödel’s idea the mathematics that describes the external reality requires a set of axioms that are beyond our comprehension. What my own research is indicating is that it may be possible to create an external reality that is generated using Pressburger arithmetic that is proven to be complete and consistent but that sentient being within that reality will create incomplete mathematical formulations.
We can develop patterns for incomprehensible realities, for example mathematics can be developed using as many spatial dimensions as you wish but they may not have meaning in our internal reality. In this way it is conceivable that we could generate mathematical expressions that represent patterns in the external reality but we could never know as we are restricted to the filter of our senses and interaction with that reality.
In pure mathematics everything is an abstract object, even mathematics itself; and as it will be seen later, this enables pure mathematics to, for example view infinitesimals as just another category of number, or for infinities to be categorised. Pure mathematicians do not have to worry about what these ideas mean ‘in reality’.
Mathematics and External Reality
So the question is, given what we understand of mathematics in describing the internal reality, what does this say about the relationship of mathematics to the external reality.
Mathematics is formulated from a set of axioms that are not provable and are subjectively chosen. This leads to ‘mathematical truth’ being limited by the axioms upon which it is based.
Mathematics is incomplete.
The fact that pure mathematics can model aspects of our internal physical reality in an incomprehensible manner seems to indicate that such mathematics is codifying patterns within the structure of external reality itself and hence in some manner the external reality is mathematical and abides to certain logical foundations.
The ability of mathematics to model the internal physical reality does not require mathematics to be an underlying mechanism that creates our internal reality. In essence this returns us back to the idea of mathematics as a reality in itself.
To understand the relationship between mathematics and reality we have also to consider the application of mathematics to physics.
Physics and Reality
“Mathematical physics gives us predictions; interpreting the physics gives us confusion”, Me.
From previous discussions, if mathematics is the codification of patterns, then physics becomes the process of matching a mathematical codification of a pattern to a perceived pattern. In other words, physics is concerned with modelling the universe that we observe and measure, and this means that physics has a direct relationship to our internal physical reality. This is a ‘classical physics’ viewpoint in which there is a level of comprehension of the mathematical patterns used to describe the internal physical reality, because those patterns tend to subscribe to our innate bivalent logic, and our intuitive view as to how the universe works.
Now on a day to day basis most theoretical physicists pay no attention to ‘reality’; all they do, as Paul Dirac once stated, is “follow the maths”.
In fact Penrose believes that most physicists, “are distinctly uncomfortable about addressing the issue of reality at all”. Having been trained as a physicist I share this discomfort. I always found it difficult to reconcile the predictive powers of physics with the fact that it told us little if anything about what made all this amazing stuff actually happen. My hero Richard Feynman had no such problem; he viewed physics as a puzzle to be solved by calculating things and he consistently and explicitly made it clear that he had no idea what in meant in terms of reality; for example in his book QED he states, ‘There are no wheels and gears beneath this analysis [QED] of Nature; if you want to understand Her, this is what you have to take’. My personal belief is that more effort should be placed on understanding how reality relates to physics and what physics can tell us about reality itself.
In the early twentieth century physicists started to explore the characteristics of imperceptible qualities, such as space, time and in particular the atom. These explorations lead to theories of relativity and more importantly for this argument, quantum mechanics. Quantum mechanics is the most successful theory of all time and although it accords with our perception of the internal reality we cannot grasp its conceptual meaning.
We cannot apply standard logic to the interpretation of quantum mechanics. As mentioned in the previous section there are topoi that generate non-bivalent logic and such logic can make sense of the seeming fuzzy aspect of quantum mechanics that appear when it is viewed from standard logic. Work by Doering and Isham has suggested that every system from the entire universe to a single electron has a characteristic mathematical signature that decides how the system will appear when viewed through a universe conformant with a specific topoi. Such ideas may in the future have a significant impact upon the way physicists view realities.
Topoi may be able to represent external reality beyond our perception of it, but our only method of assessing its accuracy is how it conforms to internal reality and there lies the problem. We view the external reality from within and quantum mechanics was the first theory to identify the importance of the fact that we are part of the system that we are modelling. Therefore physics is describing the manner in which we interpret our interaction within the external reality.
The abstraction of physics theories has not stopped with quantum mechanics, further pure mathematics such as topology, set theory and category theory have been used to develop M-theory and quantum loop gravity theory. Although these theories have been developed over many years they give us incomprehensible views of the external reality, such as one dimensional strings of energy or multi-dimensional interacting Branes that lead to a multi-verse. However none of these ideas have been experimentally verified. So we have ended up with theories that are tending toward pure mathematics and one could argue that we are back with Tegmark’s idea of the universe as a pure mathematical object.
To the untrained physicist, concepts such as energy take on a reality, but Feynman said “There is a fact, or if you wish, a law governing all natural phenomena that are known to date. There is no known exception to this law – it is exact so far as we know. The law is called the conservation of energy. It states that there is a certain quantity, which we call “energy,” that does not change in the manifold changes that nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says there is a numerical quantity which does not change when something happens”. Feynman went on to deliver the killer blow, “It (energy) is not a description of a mechanism, or anything concrete; it is a strange fact that when we calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same. It is important to realize that in physics today, we have no knowledge of what energy IS. We do not have a picture that energy comes in little blobs of a definite amount. It is not that way. It is an abstract thing in that it does not tell us the mechanism or the reason for the various formulas”.
A similar argument can be used for matter, space, time and any other fundamental characteristic you wish to mention. All of these concepts are part of our shared internal reality, whereas often physicists in the media give the impression that such ideas have some external existence, but they do not.
There is no ‘fabric to the universe’ that we can comprehend and hence little that physics can say about the mechanisms by which things happen, instead all physics can do is predict the outcome. Feynman put this misconception of physics very well when, in his book QED he said, “The more you see how strangely Nature behaves, the harder it is to make a model that explains how even the simplest phenomena actually work. So theoretical physics has given up on that”.
Other examples include the fact that much of the mathematics used in physics is that of continuous variables and it may be that at the quantum level and below external reality is discrete, and hence there may be a limitation to the ability of such mathematics to model reality. The final example of such characteristics is that of the abstract mathematical concepts of infinitesimals and infinities; once again these ideas are applied in physics and we talk of infinite space-time and singularities of infinite gravitational field, but it may be the case that infinities and infinitesimals do not exist in the external reality.
The final point I wish to raise is that physics makes little attempt to address one of the most fundamental aspects of internal reality and a potential link to external reality, and that is consciousness. There are some weak theories concerning quantum mechanical effects within the brain having a part to play but these are not main stream physics. It seems evident that, if consciousness exists then it has an intimate relationship to realities and is crucial to internal reality but no theories in physics give any indication of its origin. Most certainly if the grand theories of everything are to hold to there word then they have to explain this phenomena. The importance of addressing this issue is another reason for my research and I will show that my model can give a physical explanation of where consciousness originates.
 QED stands for quantum electrodynamics, a theory developed by Feynman for which he received the Nobel Prize.
 A Topos Foundation for Theories of Physics; book by Doering and Isham.
Physics and Complexity
Physicists tend to create theories by simplifying everything and consciously ignoring the small interactions from external sources; this is not a criticism as it is the only way that they can create the incredibly powerful and brilliant theories that we have today. However it is often these forgotten interactions that in reality cause a divergence between our predictions using a theory and our observations. Complexity science allows for the idea that our physical laws may have emerged from the complex dynamics of deeper rules that we may never be able to fathom. These deeper rules may exist in the external universe and be beyond our ability to comprehend.
Complexity science indicates that there is no justification for asserting that we can understand the deeper laws of the external reality by understanding the laws of internal reality (our physics). The idea of emergent behaviour, which has been extensively explored, means that the higher level physical laws that we see will not allow us to track backwards and understand how they were generated, this is a fact of complex behaviour and may become the final arbiter as to what we can understand about the internal and external realities.
The Physical Characteristics of Reality
From the above discussion I conclude that physics, being based upon pure mathematics and being an interpretation of a system from within that system, tells us little about external reality. However I do believe that certain observations of quantum mechanical behavior coupled with precise mathematical theory does point at potential characteristics of an external reality.
It seems to me that in some manner the external reality consists of entities that bear relationships between each other (structures), however what mediates any such relationship cannot be identified, even though we have a fairly solid model through quantum field theory. The observation of quantum entanglement supports a much more complex relationship between such entities than we can comprehend using current theory.
It is also possible that at the Plank scale the external reality is discrete in some manner and this may be supported by certain string theories and loop quantum gravity. Other research has suggested that at Plank scales symmetry breaking may lead to space and time being decoupled and discrete.
The fact that our perception of internal reality changes with scale suggests that the external reality must have structures that in some way correspond to our perceived changes in scale. That is the structures of external reality are complex.
Some may ask why I have not included the seeming true randomness of the selection of an allowed quantum mechanical state as an indication of true randomness within external reality. part of the observation process and my reason is that the model that I will describe accounts for this process without recourse to the concept of ‘randomness’. Therefore I do not see it as a potential characteristic of external reality.
In conclusion I believe that physics can only go so far in explaining what internal and external reality are and that physics is in fact morphing into pure mathematics, some of which is not experimentally verifiable and this leaves us with a purely mathematical representation of realities. This has made me question whether there is a completely different approach to understanding the fabric of external reality, one from which physics as we understand it can emerge.
 I have to say that with my hyperbolic doubting hat on I would have to question how I know that anything is correct and therefore I prefer the term ‘consistent inference’.
 A general equation is a pattern generator that when solved by substitution of numerical values for all variables gives a unique pattern.
 Penrose; “The Road to Reality”.
 Max Tegmark; ‘Shut up and Calculate’, September 25th 2007, arXiv:0709.4024v1 [Physics.pop.ph]
 Most of mathematics can be derived from the application of logic to the 9 Zermelo-Fraenkel axioms.
 This was a departure from so called propositional logic that in essence is a logic that is applied to sentences as opposed to individual objects.
 Each topos is a consistent logic formulation from which a complete mathematics can be generated. One such Topos is set theory from which our mathematics and Boolean logic can be developed. For more information see work by Andreas Doering and Chris Isham including Newscientist 14th April 2007.
 By “useful” I mean that formulations of mathematics that adopt the standard Peano arithmetic. There are forms of arithmetic, such as Pressburger arithmetic that are complete but they are very restrictive.
 In fact superstring theories use mathematics of 9 or more spatial dimensions.
 QED stands for quantum electrodynamics, a theory developed by Feynman for which he received the Nobel Prize.
 A Topos Foundation for Theories of Physics; book by Doering and Isham.
 Very Special Relativity by Andrew Cohen and Sheldon Glashow, Physical Review Letters, Vol 97.
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Metamathematical metatheorems about mathematics itself were originally differentiated from ordinary mathematical theorems in the 19th century, to focus on what was then called the foundational crisis of mathematics. Richard’s paradox (Richard 1905) concerning certain ‘definitions’ of real numbers in the English language is an example of the sort of contradictions which can easily occur if one fails to distinguish between mathematics and metamathematics.
The term “metamathematics” is sometimes used as a synonym for certain elementary parts of formal logic, including propositional logic and predicate logic.
 Structure is the relationship between entities and the recognition, observation, nature and stability of patterns.