The Origin of Theses by Charles D Norwegian. This paper is not a theory of everything, but rather concerns the fundamental origin and nature of the information Cosmos and the fundamental logic of self-reference. Even if it proves to be a false logic, it may be useful, and I have found it useful throughout most of my life, since circular arguments are almost universal and widely considered flawed. The logic presented here attempts to render the circular argument a solution, not a problem in modern thought and understanding. The aims of this paper are: - To consider the possible origin, nature and definition of the most fundamental phenomenon of existence or being in a closed, self-referentially, self-embedded, self-embodied system i.e. to consider the origin and nature of our information Cosmos. To explain and define the meaning of a closed, self-referentially, self-embedded self-embodied system. To explain that such a system is distinct from a circular argument as it has hitherto been understood, but is rather a plausible resolution to the problem presented by such arguments. To introduce new terminology in order to explain the ideas expressed. The terminology is something I try to define as early in this document as possible; hopefully in the simplest terms. I practice the argument is both simple and complex and may appears, initially at least, to be counterintuitive. For both atheists and those of mainstream Faith alike, I’ve sought to address the possible origin of being, regardless of whether it was sentient, pre-sentient or non-sentient, along with the Logos or operating principle of logic that emerges from this argument. It is not for me to evaluate the validity of the argument personally; I merely defend it as my own best insight after 44 years of formal and informal study. The model is suggestive of a Universal grammar of language; I have not elaborated on that, except to define a phenomenon we might nominally deem to be the definitive particle, and some other grammatical terms that could potentially be used to recreate a self-contained, self-consistent dictionary rooted in fundamental and pre-grammatical arithmetic terms. A closed, self-referentially, self-embedded, self-embodied system is one in which any and all of the parts exist only with reference to each other and the whole, such that neither the parts nor the whole have any independent existence, and such that the system has no external barrier or boundary surface, not even an abstract binary decision barrier of any kind. By the term binary decision barrier, I mean a barrier to define inclusion or exclusion from the system: it is widely accepted that the Universe has no such external surface. Such a system is hereafter referred to simply as a closed system, for the sake of brevity, in this document. By definition, anything not embodied in the closed system does not exist with respect to it; at least not in any semantic context. It remains plausible, however, that other such closed systems exist, but they would have no semantic consequences for each other or, by definition, they’d all be part of the same closed system. Perhaps one possible perspective is that the Universe is such a closed system. Some scholars have drawn attention to the role of the human observer. I believe each of possesses a self-realised, interactive, internal model of the universe, uniquely and intrinsically of ourselves, but based upon the same logos as the external Universe i.e. the grammar of the mind is fundamentally the same as the grammar of the Universe. These things need to be investigated further and obviously the scope of that is far beyond any single individual and is already part of the collective human enterprise of science and the many other disciplines. In Appendix 1, I show that the concept of inclusion or exclusion from set membership fails for any closed self-referential set of the kind I address. I do this by showing Russell’s paradox to be dependent upon the essentially false axioms it is framed in; its axioms make the contradiction in terms inevitable, but the profundity of this so-called paradox is a mirage. In general, paradoxes due to false axioms comprise a statement in logic and a frame-of-reference that precludes the only possible resolution that can resolve the statement without resulting in a contradiction in terms. However, using a closed self-referential system as the set, Russell’s paradox is readily dismissed. Later in the paper, I will show that the closed system cannot computationally define the whole of its own embodiment or manifestation, except in retrospect, since its number of parts increase with respect to the totality of parts embedded in, by and of it, and by qualitative change, rather than by the elapsed derivative we call time i.e. time is non-fundamental. The phenomenon we call time is, in my mind, localised and relativistic; a secondary artefact of qualitative change, or in other words a derivative, non-fundamental property of such a closed system; whereas cumulative change in the totality of the closed system, when benchmarked against its most complex location – that place or node in which the greatest amount of complexity exists - is likely deeply fundamental and could be taken to define Universal Time. The most complex object in the cosmos could be deemed to be the centre of such a time zone, and scientists are trying to create such things as conscious machines, although the ethics of this remain open to debate. I hope others will be stimulated to investigate these lines of inquiry that have perplexed me for most of my life. In the following argument, numbers are written numerically throughout to enhance clarity, except that the term unit is sometimes used to denote 1. Some symbolic lines of reasoning are itemised by lower case Roman numerals so that they can be referred to easily in the text. I will use the term datums, as the plural for some of the units I refer to in the text, to distinguish it from conventional and pre-existing meanings, contexts and uses of the term data, all of which assume inclusion or exclusion in a binary bounded system, or an infinite series of ‘Chinese dolls’ and by extension containment or non-containment in a mathematical set. My form of unary notation; is a form of simple unary arithmetic; an arithmetic simpler even than binary arithmetic: I do not use the symbols { } of unary set notation, to enclose the internal set or sets of phenomena: that would imply a container unit datum that was fundamentally, and qualitatively different to the contents. It would also define a decision barrier with reference to which the contents could be classed as included or excluded, even as an abstraction, from the system; something which cannot be justified for the reasons previously stated. Embodiment implies neither inclusion nor exclusion; therefore neither containment nor non-containment: the Universal set is a closed system that is self-referential, and that both embeds and embodies itself but does not contain itself. The use of curved brackets can be justified for visual clarity however, as will be seen later: I have had to use them as there is no conceivable simpler alternative that aids the understanding. For those who must know at the outset, what the brackets really represent is loosely analogous to the partial, cross-sectional, surface arcs of spherical bubbles, although in a 2-dimensional document all that can be depicted are circular arc sections, hence my use of pairs of brackets as the best symbols to approximate one of these bubbles, except in one special case, where no concentric containment exists, a philosophically crucial case. These arcs do not imply binary bounded containment with any possibility of exclusion from the system however; rather they represent self-referentially, self-embedded, self-embodied interrelationships alone and they rightly show that surfaces within that finite system are curved: these are purely qualitative phenomena in the simplest state or origin of the Universal set. The quality of Being would seem to go hand in hand with provident pre-sentience when it comprises the Universal set; this I leave for others to debate but I posit that the logic of being itself is the very logic of provident consciousness, narrative, evolution, complexity and all things upon which the meaningful cosmos depends. It appears something vastly simpler than the intelligence commonly assumed to be its prerequisite, but even the Bible says that first there was the Word or Logos, the operating principle of the cosmos. So it is this that we need to formulate and it is this that I am trying to help with in this paper. Some apparent contradictions emerge in my argument that will be resolved as it progresses; particularly the one concerning the nature of the sub-semantic unit datum, and the fact that it appears and may well be, according to context, the very same ‘object’ that any 3 sub-semantic unit datums themselves make manifest when self-referentially embedded and embodied in themselves: a 4th sub-semantic unit datum forming the system’s 1-dimensional surface inevitably emerges from the 3 and consists of the difference manifest and distributed among the other 3 bits: the difference between every and any two and any remaining one. Consider 3 identical, irreducible units of phenomenology, or ‘information’, in a Universal set or closed system containing only those 3 unit datums and a hypothetical 4th identical container unit datum: I have already explained that this 4th unit datum emerges inevitably from the 3, but this will later become clearer still. The 4th unit defines the surface of the system that emerges from the 3 unit datums. Each of the 3 unit datums may be called closed, self-referential, self-embedded, self-embodied unit ‘bits’ if we choose the full informational descriptors , but I will call them sub-semantic unit datums for simplicity. I also describe them as sub-semantic unit datums, because any one of them, taken on its own as a discrete item, has, nominally at least, no intrinsic semantic value, in any manifest respect, except in retrospect, or by reference to a something that does, and the latter requires multiples of such sub-semantic datums, as I will show. There is one, and only one, phenomenologically unique difference embodied within the preceding system: the difference between each and any 2 sub-semantic unit datums and any remaining sub-semantic unit datum. This difference is unit quantitative quality, the quantity being unit, or 1, and the quality being the fact of the presence of a unique difference. In other words there is unit quantity of unit quality of difference embodied in the system; moreover, a quantity of any quality is a magnitude; thus we have a system with a unit magnitude, indeed unit fundamental magnitude as will become clear: however it is much more profound than that alone, as will also become evident. I am going to show that these particular units of magnitude, quantity and quality are fundamental and show that the fundamental unit of quality is a 3-dimensional spherical qualitative spatial ‘volume’, whilst the fundamental unit quantity is the 4-dimensional spherical qualitative surface ‘area’ that both defines and includes the fundamental unit volume in the simplest closed system. Note that whilst the surface area is 4-dimensional, in magnitude, the surface itself is 1-dimensional in quality, and contains the 3-dimensions of volume. Likewise, whilst the spatial volume is 3-dimentional, in magnitude, the space itself is 0-dimensional and purely qualitative in nature. Thus volume as used here does not refer to everyday space, but to something more fundamental, qualitative or sub-semantic space. Note that the difference concept, that between the sub-semantic unit datums, just described, appears to be merely its own manifestation of itself, that is to say, self-evident; but the concept of self-evidentiality is neutralised if not dispelled by my full argument; my the logic rests on no assumptions I can identify as such; at least none that cannot be dispelled in retrospect, and its only ‘virtual’ axioms are derived from first principles even where they agree with pre-existing ones. My full argument attempts to precisely define self-evidentiality. It does not matter whether this difference concept is actually manifested; what matters is that it is both possible, and potentially able to be manifest, and that its manifestation would be distinguishable from its mere possibility. Thus it self-evidently possesses and may rightly be deemed to be unit existential semantic potentiality. The only internally ‘observable’ structure within the system is this unit existential difference; the 3 component unit datums of which it is made are not internally observable, but we can retrospectively infer that they must nonetheless be present and thus exist, since if they were not there, and did not exist, the unit existential semantic potentiality would not be there either, leaving the system devoid of any self-existent semantic. It follows that this 3 unit sub-semantic datum closed system, and the 4th unit sub-semantic datum it inevitably manifests, is the unit fundamental existential semantic potentiality, and that it is also therefore: - The fundamental unit existential difference operator. The fundamental unit existential difference operand. The fundamental unit of all observable existence in a closed system. Later in my argument I denote this phenomenological entity by the symbol -0, but this is not to be read as, or confused with minus zero. The continuing argument attempts to defend this proposition further, eventually extending it to larger systems, and speculating about various other things. Conclusions 1: - In the simplest possible self-existent, closed self-referentially self-embedded, self-embodied system comprising 3 sub-semantic datums and a 4th potential sub-semantic datum that is the potential difference between any two sub-semantic datums and the remaining unit sub-semantic datum, there is unit fundamental existential semantic potentiality denoted by -0. This -0 is in itself the potential difference between any two sub-semantic datums and the remaining unit sub-semantic datum irrespective of whether that difference is manifested or observable in any way. Thus -0 is the objective definition of self-evidentiality; something generally deemed subjective, and equally the subjective definition of self-evident; something hitherto deemed objective. The binary distinction between the objective and the subjective breaks down under this Logos, to be replaced by the transcendentally qualitative. The word transcendental may be unpopular among scientists, but it is appropriate I believe in this instance. A discrete unit sub-semantic datum is a closed self-referentially self-embedded, self-embodied system but it has no semantic potentiality. However, it has sub-semantic potentiality because it is not constrained by any frame-of-reference. However, 2 sub-semantic datums cannot exist without a 3rd sub-semantic datum to define their difference. That difference requires a 3rd bit or unit sub-semantic datum. The triplet would possess a rotational symmetry and a twist, if each bit were indistinguishable, and in addition a 4th bit, the semantic potentiality: the difference between any 2 and the remaining unit bit would inevitably be present as already explained: this potentiality -0 is emergent and implicit. The amount of bits seems to ‘want’ to explode like a chain reaction or exponential curve and indeed we believe that the observable, and indeed inferable, Universe began in such a way. Let us consider 3 unit sub-semantic datums, any one of them defining the difference and the relationship between the remaining two, and all of them being equal and identical although different in the sense that they are each discrete: - ()()() (()()) ((())) The third of these configurations, ((())) is in fact identical to the first ()()(): because there are no discrete unit sub-semantic datums they cannot contain each other; if they did there’d be no unit sub-semantic datum to define their discreteness. The second option, (()()), depicts the fact that any two unit sub-semantic datums can be defined by reference to a third that defines the difference between itself and the remaining two i.e. the discreteness of any and/or all. But this is bilaterally symmetrical and neither of the ‘contained’ bits can define a frame-of-reference that ensures one is not the other. This system has only 3 sub-semantic bits but it does not have the potential to break its bilateral symmetry: it must have semantic potentiality therefore, and it is its symmetry that confers that property. For instance, although the two mirror-imaged bits are indistinguishable, the third bit makes them both different, and the symmetry is the unique feature. In reality, there need be no bilateral symmetry at all: the triplet may exist as a kind of Mobius loop with indeterminate orientation with no external reference points. But in order to comprehend it at all, it is necessary to use models that more closely reflect everyday concepts: this enables meaningful observations to be made. If we disregard the container bit things become even clearer: - (()) and ()() In the first case, we have what appears to be one sub-semantic datum containing another, but no 3rd datum to render them discrete, and since they are not discrete, neither can contain either. In the second case, we have a totally symmetric arrangement; they would be discrete with respect to each other and to their unit sub-semantic datum container, but neither can identify itself as unique with respect to the other by any potential state change the system allows i.e. by any semantic potentiality the system permits. However, 2 sub-semantic datums in a 3rd sub-semantic datum container possess absolute symmetry and a static semantic: this system has no semantic potentiality. However, it possesses absolute symmetry in static semantic even though it lacks semantic potentiality. It possesses the single most fundamental quality of self-existence and of any and all possible existences: absolute symmetry in static semantic. However, it does not possess any internally defined magnitudes because no unique differences exist in such a system that could enable any magnitude or magnitudes to be defined. Indeed such objects, or phenomena, could well exist, or have existed, and may even play a key part in the origin of any closed system, such as the Universe or Multiverse, if such phenomena are closed systems: which of course is one of the key issues. A unit sub-semantic datum in a unit sub-semantic datum container in a closed system is a logical contradiction in terms: it possesses no semantic at all in any closed, self-referential context of any kind, as I have explained: it cannot even possess concentric containment. Does this mean that a single unit sub-semantic datum, in discrete isolation i.e. not contained in any closed, self-referential system, defines non-existence and is there such a thing as non-existence in absolute terms: indeed is there such a thing as absolute terms either outside or inside any closed, self-referential system? My intuition is that the answer is yes, if only nominally: non-existence is in practically definable terms unit sub-semantic datum in isolation, but I will not elucidate fully it all until later. The contradiction is that non-existence has therefore a definable existence, at least beyond time, if only as an abstraction. A single unit sub-semantic datum, in discrete isolation i.e. not contained in any closed, self-referential system, may nonetheless be treated as existent outside any closed, self-referential system; i.e. in an open-existence. It can, in a retrospective sense, be semantically defined as indistinguishable from unit sub-semantic datum in any closed, self-referential system and it may therefore be deemed non-semantic in discrete isolation i.e. outside any and all closed, self-referential systems, but it is only non-semantic in closed, self-referential terms even though we are referring to it as an datum beyond the close, self-referential realm. Even unit sub-semantic datum in discrete isolation outside any and all closed, self-referential systems, is a semantic object in the absolute sense, by virtue of the very fact that we can even define it, if nothing else, as non-semantic: it is semantically non-semantic, but only with respect to any closed, self-referential system it is non-semantic or observationally meaningless, so the apparent contradiction in terms, is resolved, for the purposes of anything or anyone, contained in a closed, self-referential system. However, I will return this matter again later, as this doesn’t resolve it from all possible perspectives: we have not established what this ‘object’ is, of, by, and in itself, when it is not contained in a closed, self-referential system, and we cannot rule that possibility out: should it be less than impossible it would be as good as certain that such ‘objects’ exist and perhaps were, or are, the origin of all things. If we can infer that they do, then we can infer that other discrete closed, self-referential systems exist as well, including other Universes. Moreover, this logic does no rule out the possibility that they are all interconnected in a Multiverse, neither the possibility of a multi-world scenario, things I’ll address later in this argument. The example ((()())), is of course the 3 sub-semantic datums with a 4th emergent ‘container’ or more strictly difference bit, that I have shown is the simplest system expressing -0. The containment is an artefact of the notation. Conclusions II: - A closed, self-referential system comprising 2 sub-semantic datums in a 3rd sub-semantic datum container possesses absolute symmetry in static semantic but lacks semantic potentiality. It possesses the single most fundamental quality of self-existence and of any and all possible existences: absolute symmetry in static semantic. However, it does not possess any internally defined magnitudes because no unique differences exist in such a system that could enable any magnitude or magnitudes to be defined. I will call this system the unit fundamental quality in absolute symmetric in static semantic: I will denote it by: 3/4(-0). The first meaningful phenomenon that could ever have existed anywhere in the whole of the totality of existence that includes our Universe, its origin and causal descent, the origin and causal descent of all other phenomena contained in that totality, and the origin of all non-causal phenomena contained in that totality, must have been the object just defined, 3/4(-0), a closed, self-referential system comprising 2 sub-semantic datums in a 3rd sub-semantic datum container, if there was an Origin or Creation of any kind. The first phenomenon to descend via causality from 3/4(-0) may not have been a Universe; it may have been a precursor to the Universe, Universes of Multiverse; whilst I remain impartial, I think the possibility should be considered. Who is to say as yet, what it may have developed into, or that such a computational system could not have evolved in many things before any Universe or Universes arose? There is scope here for multiple religious views, including novel ones, to be accommodated. The first thing to have created itself might even have been God, depending of course, on how we define or understand that word. That this closed, self-referential system comprising 3 sub-semantic datums and a 3rd sub-semantic datum – the bit that defines the difference between any two of the others and the remaining unit sub-semantic datum – had no semantic potentiality does not imply it could not have been the origin of semantic potentiality or more strictly -0, the next most complex phenomenon, for instance by undergoing an internal division. Also… Any causality that could arise in any closed, self-referential system, requires the emergence of frames-of-reference. Moreover… The simplest object containing a frame-of-reference is -0. And… For -0, the frame is most simply considered as the spherical surface area of the spherical volume it defines. And… Embodied in, manifested by, and created of -0, is a frame-of-reference that defines a quantitative quality; 3 sub-semantic quantities of unit sub-semantic datum or being, a quality, and the quantity that emerges from the difference between any two and the remaining one. This concept of quantitative quality is crucial to my reasoning. -0 possesses a unit magnitude of quantitative quality. -0 embodies its parts, both the quantity and the quality - the surface and the volume analogues respectively, as will be seen later - in a single closed, self-referential system. Any closed self-referential system of any size, when measured in units of -0 and not in units of sub-semantic datums, may be deemed to have a nominal magnitude of 1, with respect to anything the system contains, since it’s magnitude of being, the quantitative quality it defines, is 1 whole closed, self-referential system: however, it may contain any number of sub-semantic unit datums and any multiple of -0. Any closed, self-referential system will thus define its magnitude of being as 1 in any event, by virtue of its very definition. In a religious or spiritual sense, I see a closed, self-referential system as consistent with One Being in many states of being. While many Western secular thinkers perceive such language as sentimental, subjective nonsense, such prejudice or bias serves selves rather than callings; misunderstanding rather than truth. Considering that the whole is greater than the sum of the parts, to use the old cliché, and that the whole of the Earth is far greater than its parts, if not nearly as great as it ought to be, then is it really implausible to view the whole of the Universe or the whole of the rest of existence that may lie beyond it, as no greater still; not even great enough to be justifiably equated with God or at least Heaven in the making? I am not prepared to close my own mind to such interpretations, given that anything possible, regardless of however improbable it may be, is now considered to be as good as certain by some thinkers, and also given that the whole of existence may well be infinite, making even the most improbable thing absolutely certain. I choose to be as diplomatic as possible but affirm the eusocial and spiritual super-organism of the Family of the Universal Faith. The origin of causality, i.e. causal logic and its effects, together with the conservation of information, including semantic information, and the definition of different kinds of information in precise terms, is something I’ll address in more detail later. However, any meaningful i.e. semantic content in a closed, self-referential system must be capable of being contained within, or embodied by, that system and defined by its state, and defined by reference to any constraints on that state and the historic events that led to it. Moreover, the question of how any closed, self-referential system, such as the Universe, Multiverse or any other such system, can nonetheless have no external surface must be addressed: that is the popular misconception of cosmological containment must be addressed: - The cumulative surface area of the total -0’s contained in the system is equal to the total surface area of the observable spherical event horizon with respect to any definable position in the system and this area is measured in multiples of unit 1/4(-0) — the boundary bit sub-semantic datum cannot be observed — but the position of the observable spherical event horizon changes with the unit of -0 deemed to be its centre and thereby with the position of any observer, such that: any unit A on the observable spherical event horizon with respect to which any unit B was the centre, would find itself a single unit B on the observable spherical event horizon with respect to which unit C was the centre. The total number of unit multiples of -0 contained in the system would determine the surface area of the observable spherical event horizon. In fact, the -0 content of the whole system may increment with the odd number the sub-semantic datums it embodies; increments that result in an even number of sub-semantic datums, do not increment the quantity of -0 contained in the system. Prime number increments may have special significance. The conservation of the meaningful meaning of meaningfulness, as I see it, in a closed, self-referential system is something I’ll address later. It is possible to conserve random data without conserving any kind of meaning, but random data is not semantic information. The interesting thing is that new sub-semantic data doesn’t seem to random either; it seems to be perfectly ordered and some speculate that the Universe is still creating it, as do I: I wonder if it is the true nature of dark matter with dark energy balancing the books as it were by generating randomised data space; the space we call empty space but now know isn’t in fact empty. The phenomena so far considered do not require causality i.e. cause and effect, or laws. In the absence of any internal frame-of-reference, a system would not be constrained by any laws of causality or derivative formal logic. Thus, it could do anything and everything, without regard to cause or effect, until such constraints arose within it; presumably once certain milestones of complexity, asymmetry and frames of reference are reached, some of which appear to be known and proven, such as those implied by the limits to the dimensionality of the algebraic logic systems consistent with logical causality. The issues are a matter for mathematicians and physicists. A single unit sub-semantic datum, in discrete isolation i.e. not contained in any closed, self-referential system, may nonetheless be treated as existent outside any closed, self-referential system; i.e. in an open-existence. It can be semantically defined as absolute unit sub-semantic datum. Even non-existence can be deemed with respect to any extant closed system to be absolute unit sub-semantic datum. Conclusions III The fundamental unit area if you prefer, is the fundamental unit of quantitative quality, and is -0; strictly, the unit fundamental existential semantic potentiality in closed self-reference; and is 4-dimensional; thus, all areas, in a closed, self-referential system are 4-dimensional and thereby more complex than volume, which is 3-dimensional as we have seen; indeed just as appears in everyday reality. This is because the fundamental unit volume, is the fundamental unit of quality, is 3/4(-0); and is 3-dimensional, as explained. An area without a volume cannot exist, since without a volume it would have no quality to quantify, no volume to contain. No surface could be infinitely small or it could contain no volume; it would be a quantity without quality, even any undefinable quality. No volume could be infinitely small or could not be definable by a surface area. It would a quality of no quantity; making it As I have explained, -0 is the simplest possible self-existent closed, self-referential system embodying semantic potentiality, which is by definition what -0 is. This suggested modulo arithmetic might be a useful means of further exploration, in particular mod 3 arithmetic, and I have used a kind of unary set notation to symbolise my argument as well, as its the simplest way I know of representing numbers, or more correctly in this case, units of existence or pure data. Using a unary notation to depict a sub-semantic datum as (), a closed system with 2 sub-semantic datums in a 3rd sub-semantic datum container looks like this: - (()()) ((())) this variant by the way, is a meaningless artefact of the notation since the two contained unit datums would need a 3rd to render each discrete giving: - ((()())) which of course is 3 sub-semantic datums in a 4th container, and as we have proved is the simplest system expressing -0. ()()() 3 unit data in separate systems; not in closed self-reference. i (())() 2 unit data in one self-referential system; the 3rd outside that system. ii ((())) 3 unit data in closed self-reference. iii (()()) 3 unit data in closed self-reference. Iv iii and iv embody the potential for a unique internal ‘binary’ difference in states; although that potential is only realised by iv: being embodied by the fact of there being two possible alternative ‘binary’ states, namely iii and iv: the difference between any 2-bits of unit data and any remaining bit AND the difference between this and any other arrangement that manifests or expresses no such difference (the difference explained at the outset), as is the case in instance iii. However, to have any meaning external to itself, the 3-unit datum object has in turn to be embedded in a 4th unit datum as follows: - (((()))) 3 unit embedded in a 4th unit datum in closed self-reference. v ((()())) 3 unit embedded in a 4th unit datum in closed self-reference. vi Once the 3 unit data are embedded in this 4th unit datum, which effectively functions as the largest frame of reference or boundary of this cosmos, an interesting quality arises: an additional alternative internal state emerges, as follows: - ((())()) 3 unit embedded unit data in closed self-reference. vii System states, v, vi and vii comprise the fundamental operand/operator, embedded in a frame of reference consisting of a single additional 4th unit datum. The mod 3 operation can be used to do modulo arithmetic. It can be written, using the unary notation, perhaps idiosyncratically, as: - |mod ((())())| …or mod |(())()| if we disregard the containing frame, but whilst this may be easier on the eye, I consider that it would be incorrect. This is a special use of the unary set notation where |mod x| means the transcendental remainder on division by x: we cannot speak of qualities or quantities in this system, because it is too simple to define any such concepts. |mod ((())())| can be expressed alternatively, in linguistic terms, as: - |mod -0| … This is what this arithmetic really is. I use the magnitude symbol around any value x, as in |x| for instance, to indicate not that we are dealing with actual magnitude, nor even any signed or unsigned arithmetic, but to indicate that we are dealing with a fundamentally pre-qualitative, pre-quantitative, transcendental arithmetic in which neither quality nor magnitude have yet to arise or be defined. This is because there are no frames of reference yet to define any dimensions that in turn could create or define any such descriptors. Nonetheless, there is no doubt we are dealing with a semantic phenomenon; namely an observable internal, and self-existent difference within a closed self-referential system, whereas the components of this difference are sub-semantic. On the left hand side in the following description, is a unary set notation denoting 3 sub-semantic datums. However, as explained previously these are embedded in a system consisting of a single additional 4th unit datum: they are arranged differently to distinguish their possible internal relationships or states, but all consist of the same parts: 3 units of sub-semantic unit data. These 3 unit sub-semantic unit datums are sufficient to manifest -0 as described previously. On the right hand side is a decimal representation of the unary notation and I will explain why I have chosen thee numbers later in the argument. a = (((()))) decimal 0 viii b = ((()())) decimal 1 ix c = ((())()) decimal 2 x State vii is decimal 0 because it is indistinguishable, from non-existence; embodying no internal differences of, by, or in itself, and manifesting no discrete parts. State ix is decimal 1 because it manifests 3/4(-0), the included qualitative partial of -0. Finally, state x is decimal 2 because it manifests and embodies both the qualities of ix, the included qualitative partial 3/4(-0), and also -0, which defines the unit quantity of that qualitative partial. Stripping away the external 4th bit for a moment to make this easier on the eyes; the container system leaves: - a = ((())) xi b = (()()) xii c = (())() xiii I will make some repetition here: - State xiii is the unique state; the single qualitative difference between any two unit data and any remaining unit datum, expressed in a structural form that enables such a distinction to manifest. So referring back to vii, ix and x, the unique state is x: this object is the embedded unit existential difference operator/operand in closed self-reference: embedded that is, in a 4th unit datum, the container system. Any two bits can be distinguished from the remaining one, in reference to the enclosing 4th bit container system; and in fact there are two possible alternative ‘binary’ states: the difference between any 2 unit data and the remaining bit and the difference between this arrangement and any other arrangement that manifests no such difference, as previously explained. State xii is not unique, because it contains a symmetry that fails to distinguish between either of the two centrally enclosed bits, as the notation shows, and the 3rd bit simply functions as an additional encapsulation inside the 4th. However, xii does indicate that there are 2-bits that have an existence with respect to each other and both are embedded in the while 4-bit system. Thus, although there is no way to uniquely identify them, the fact of their mirror symmetry and non-unique difference is real within the system. They can therefore be equated with decimal 1 for simplicity. So referring back to ix, this object is a unary pair in which no unique state can be identified because neither bit can be distinguished from the other, and the enclosing bits cannot be distinguished either despite the illusions suggested by the notation. On the other hand, in the case of state x, any two bits can indeed be distinguished from the remaining one, with reference to the enclosing container system; and in fact there are two possible alternative ‘binary’ states: the difference between any 2 unit data and any remaining unit bit AND the difference between this arrangement and any other arrangement that manifests no such difference, as previously explained. My apologies for the repetition, but the second set of notations are easier on the eye, having dispensed with the 4th container bit. State xi has no internal differences at all; none of the bits can be distinguished from each other and furthermore, no symmetry or structure can be distinguished. Although the notation gives the illusion of structure, the apparent spherical concentricity is merely an artefact of the notation. This object is indistinguishable from non-existence embedded in itself times 4; it would make no difference if it were embedded in itself ad infinitum: it would not know itself from any part of itself let alone its whole, and neither would anyone or anything else. By definition let |b mod a| = the transcendental remainder of b upon division by a. …we’re only dealing with magnitudes but strictly we have no definition for magnitude yet, but we’ll have one by the end of this argument. Incidentally, -0 may be equated with a quantitative difference the quality embedded in its component or partial 3/4(-0). The simple closed system under consideration has yet to create a reality to which such terms as magnitude and quality have any meaning: thus it is a transcendental system of pure, abstract data, for want of any description. In reference to viii, ix and x (or xi, xii and xiii) let: - a = |0 mod 3| = 0 viii (alternatively) b = |1 mod 3| = 1 ix (alternatively) c = |2 mod 3| = 2 x (alternatively) d = |3 mod 3| = 0 … where: d≡a These are the same statements as viii, ix, x (or xi, xii and xiii) in an alternative form that enables us to bring the modular arithmetic into the argument, I think more clearly and at least in a form easier on the eye, for the purposed of what follows, although I will certainly show later how it looks in the original modified unary set notation. A closed, self-referential system must nominally always contain unit information, i.e. 1, with respect to its whole, wholeness being oneness, and zero information with respect to anything not contained in itself: I equate wholeness with 1, which seems reasonable, since any parts can be treated as, well, parts of the whole. As for anything outside the system, it does not exist by definition. Therefore: - |(a - b - c)+(b - a - c)+(c - a - b) mod 3| = 0 with respect the outside which by definition does not exist in any meaningful terms. xiv |(a - b - c)+(b - a - c)+(c - a - b) mod 3| = -0 with respect the inside. xv … these are both |-3 mod 3| on the left hand side. So we have: - |-3 mod 3| = -0 From within this closed, self-referential system, |-3 mod 3|, which is the arithmetic sum represented by the preceding symbolic letters and sums: xii, xiv and xv, cannot be treated as equal to a conventional zero. This arithmetic will not allow that, given the closed, self-referential context and the whole drive of this philosophy: this is NOT a case of: - |-3/3|= 1 remainder 0, or in other terms |-3 mod 3| = 0 It is neither of these. This is because what appears to be a minus sign is in fact not; it is actually the -0 as defined previously: it is emphatically NOT the minus operator at all, which this simple system has no frame to even define. The conventional minus operator and all the other operators and operands can be derived from -0 but it much more fundamental that any of those conceptual derivatives. By extension |n(-3)mod3| = n(-0) = unit totality, and the rest of the argument still applies. Since any observer must be contained in the system in order to observe it, we may ignore the zero option xiv, nothing lies outside the system, and so we’ll consider the -0 option xv. Let d, e and f represent 3 existential difference operators/operands. Although the system contains 3 -0, we can generalise and treat the totality of data in any closed system containing -0 or more, including the simplest system that embodies -0, as nominally equal to 1 (one whole system) for the following part of the argument; a more conventional way of denoting the whole of something, but we could equally have chosen to call it x, any multiple of -0. Thus, for a closed system containing any number of -0 as previously explained and so we have: - d – e – f = -1/3 xvi e – d – f = -1/3 xvii f – d – e = -1/3 xxviii …Given that all -0 are indistinguishable under any conception of their nature; just as their sub-semantic datum are indistinguishable. d, e and f are identical, equal but different: they are three existential difference operators, but they are discrete objects: they are identical for all descriptive purposes; they are equal in any definable or measurable quality of themselves with respect to the system, each other or themselves, if they even possess any definable or measurable quality at all, and yet they are different by definition: so in linguistic terms we may express this as: - d = identically different equality OR d = identically identical identity (x y z) e = differently equal identity OR e = identity identical identically (z y x) f = equally identical difference OR f = identically identity identical. (x z y) Now let’s do this: - G = d – e – f = |x – y – z| - |x - z – y| - |z – y – x| = x + y – z H = e – f – d = |x - z – y| - |z – y – x| - |x – y – z| = z + x – y I = f – e – d = |z – y – x| - |x - z – y| - |x – y – z| = y + z – x G + H + I = x + y + z This object is irreducible and indivisible in terms of its own self. All of the preceding three G, H and I are the same in all respects but for the fact that they are discrete, by definition. Hence: - |(f - d - e) |+|(d - f - e)|+|(e - f - d)| mod identically different equality = -0 xix G H I Thus x + y + z = -0 xx In other words, the intersection between mathematical and linguistic logic, and the basis of all pure observable information within an observable cosmos that is a closed, self-referential system could be defined as the identically equal difference; or the differently identical equality; or the equally different identity: this is what -0, which may called the existential difference operator/operand, really is: it is the unit observable datum with respect to any observer contained within a closed, self-referential reality. The set of all things that exist with respect to us, in our cosmos, is by definition a closed, self-referential system; otherwise it would not be everything that exists. It’s conceivable that most of existence does not in fact exist with respect to us; certainly the past does not exist anymore, even though we can be certain it once existed and was very real. However, the past has left its traces everywhere as information is conserved within the Universe. But, other spatial existences, universes, may be completed isolated from our closed, self-referential system, leaving them beyond all observation and in the realms of speculative inference. Using the admittedly hard-on-the-eye but more fundamental unary notation we have: - ||((()()) ())+(((()()))))+((((()))))| / mod((()()) ())| = -0 xx ||((()()) ())+(((()()))))+((((()))))| / mod(((()())))| = -0 xxi ||((()()) ())+(((()()))))+((((()))))| / mod((((()))))| = -0 xxiii .. These three are equivalent to the expression on line xix, but in unary arithmetic, showing that the linguistic version and the mathematical version are authentically the same in semantic terms. (The brackets may be conceived approximately as the cross-sectional surfaces of bubbles, with some limitations of course; these can also be treated as illusory artefacts of the notation as already mentioned.) Thus, the object that divided by any meaningful descriptor or measure of itself, whilst being and remaining equal to itself and to the sole difference it contains or embodies, and that is itself also the result of any such operation, is the fundamental operand and also the fundamental operator: it is the unit observable datum and unit semantic potential difference -0 and it is the definitive particle when expressed linguistically. (c) 2010, 2011, 2012, Paul Trueman
Posted on: Sat, 25 Oct 2014 09:38:01 +0000