Discourse on the Empty Set

In the article 'Ontological Logic' I addressed the issue of totality being analogous to the empty set. If one doubts totality and oneness as actually existing, then it would follow that the concepts of 'singularity' and 'conception' themselves would fail to acquire meaning. If one may recall the article 'Metaphysics of Awareness', I addressed the issue of nothingness; I said that if the human mind is to be aware of ‘nothingness’ then it is to be treated as a category of existence. This of course contradicts our conventional understanding of nothingness, so perhaps it is that our conventional understanding should be re-examined? From a similar perspective conventional understanding of emptiness would again fail us in this case, as I shall expound below.

The heart of the matter and what may seem contradictory at first glance is that emptiness is a totality. The concept of a singularity pertains to an abstracted totality; to isolate an instance of this totality we would use an abstracted form, what we call an aspect of generalisation. One particular aspect of generalisation may be emptiness; the argument is that conception of emptiness produces emptiness itself, which would not result in a higher ontological state. In a similar fashion it could be said that a totality of totality does not produce a higher ontological state, but the process is used in establishing an ontological circle. The isomorphic nature of these two arguments is no coincidence; emptiness can be used to establish an ontological circle in itself which should then imply that emptiness is an ontology in itself, which I shall now show.

The reason I justify using the empty set as direct analogy is that an idea of 'emptiness' in itself is not truly 'empty', in its becoming an idea. This is analogous to the argument on the nature of nothingness. From a mathematical perspective it is possible to isolate that which 'is really empty' by adding a basic structure to it. It is in this basic structure that the totality arises, for we can now meaningfully speak of the emptiness to ascertain its mode of existence. 'Emptiness' can now be treated as a category of existence, a property which can be assigned to every element of being. The empty set must be compared to every possible element to ensure that it really is 'empty'; the empty set is consequently the absolute totality of the emptiness of every element. It is by virtue of this ontological totality that the ordinal hierarchy may be constructed, in taking the set of the empty set, which I would call 'conceiving' it, a greater structure is produced. What the empty set really represents then is a basic set structure, moreover its singularity, and the abstract intuitive 'emptiness' becomes a complete totality in itself.

The argument is an illustration of ontology as being the foundation of being, to criticise ontology would occur upon its very grounds of being, the criticism itself would contain an ontology which would establish its self-contradictory nature.

I will also address the idea of conception of being an 'identity' operator, that which gives us the element of conception being conceived upon back untouched. In its operator status there is a structure holding the category of the 'identity characteristic', as such a category of existence is formulated. Each operator must be compared to this category to ensure the identity operator leaves the operand unchanged, thus the operator could be said to be a totality of the identity characteristic. Therefore if we are to call conception an identity operator then we must concede that it will nonetheless be a totality in itself, and this totality allows us to assert its meaningful nature via the Fundamental Ontology.

The Practical Application of Ontological Logic

I explained in my previous article that the purpose of logic is in equating concepts with their singularities. It is overdue for me to explain that any substance in itself can be conceived as singularity, we may have concepts referring to a physical objects and thus they must too be considered singularities. Any form conceivable is a singularity in itself; in the case of a physical object the forms of the singularity would be every particular instance in which it may appear. If one studies my axioms they will understand that this now entails that all substances, what could otherwise be called any element of existence the mind is aware of, obeys a singularity structure; this is an elaboration of the idea of existence being shaped by the singularity-form archetypes, though it should not be a surprising consequence when one considers ontology.

The process of naming singularities occurs a posteriori. Different individuals may call the same thing by different names, if we may temporarily ignore the problematic nature of asserting that different individuals may refer to the 'same thing'. The only a priori knowledge is of the Ontological Circles whose existence shapes our very perception; they are the lenses of awareness which give rise to experience, and as such they must exist in the human mind before experience occurs. This is summarised as below:

As we can see what we are attempting to study is the faculties of reason and logic, of course through their very being, which is what entails Ontological Logic. As hinted to before, this is the highest domain that the faculties should concern themselves with, for the Soul substance of the human Will is completely beyond the measure of words and thoughts as eluded to many a time. 

As a mathematician I am interested in complete abstraction and totality, which gives rise to a sense of aesthetic beauty. A reader of this philosophy should ask the question as to why I have chosen four ontological circles, or rather why four have arisen over the course of the extended discourse. There are four because I deemed that to provide a clear enough elucidation into the structure of the faculties of reason and logic, of course the trend of my philosophy is to lead us unto oneness. There is only one ontological circle, as there is only one mind and one self. Awareness, singularity, conception, these are branches of the same tree viewed from different angles. The tree of being. In an alternative system of philosophy there may be five, six, or 10,000 ontological circles. But my philosophy transcends its being as a system, it studies how philosophical thought arises in the human mind. The heart of the matter is the issue of an ontological circle itself, not the particular ones. 

In this article I am providing a discourse on the relation of reason to our experience of the world. Clearly, not all is absolute oneness. From the disintegration of oneness into form, and that form again in a hierarchy of being, our perception of the known world is as such constructed. We can meaningfully speak of every branch of this tree aside from its crown. It is clear of how the forms of one singularity may be individually named by concepts, but this does not account for the mingling and compounding of singularities to produce the objects of our perception.

To make this more clear let us proceed by way of an example. It can be said that the sky is blue. When we speak of the singularity of the sky we do not refer to its physical properties, but to the teleology of the substance, what the mind perceives a sky as constituting defined by the purpose ascertained by conscious awareness. There is also the singularity of blueness. To prescribe the property of blueness to the sky is to say that the singularity of the sky is a form of the singularity of blueness. Of course we could equivalently say that the singularity of blueness is a form of the singularity of the sky, for in this case the category of blueness is perceived as being fundamental to what makes the sky, the sky, in this particular instance. During the night we may say the sky is black and full of stars, for a more universal definition of the sky we would need to abandon attachment to the essence that changes and analyse its fundamental constituent being. I suppose that would entail the singularity to be the scientific definition of the sky as an atmosphere of a planet. 

In general a singularity A can be said to take on the property of a singularity B if one partakes in being the form of the other and vice versa. 

Substances can also be perceived to be the compound of two different singularities. A bird could be considered a singularity in itself, but this singularity could be said to be composed of the constituents of the bird such as its wings, feathers, and so on. Each of these singularities is amalgamated to produce the singularity of the bird. This is a generalisation of the idea of properties, to explain how a perceived singularity arises in the conscious mind.

This presents a rather complicated universe of the experienced world consisting of singularities and forms each participating in the others existence. Hopefully I have presented a sufficient detail of the precise mechanism by which this experiential construction takes place in the domain of human perception. The names, concepts and ideas of this world are transient and impermanent; forms will always be forms but what form they are and in what singularity is the shifting chaos of this world. Only absolute Oneness stands eternal and timeless, it is only absolute Oneness that is real. It is such that the practical applications of Ontological Logic are not of the greatest concern to me, I simply wish to study their theoretical detail. Ontological Logic itself is the birth of reason and logic, it is the faculty in itself and of itself. 

Ontological Logic

Logic is an area which is one of my greatest joys in studying as a Mathematician, but in presenting my own theory detailing its foundations to the inner cosmos which it projects, is no other than a monumental task which brings me to the limits of my mental capability. The true difficulty of the task lies in the fact that when producing a critique on logic, we must use the rational faculties of the mind to produce it in the first place. To hold this level of introspection and mindfulness and most importantly to bring it from the esoteric darkness into the knowable light is one of the most strenuous yet rewarding tasks a human can undertake.

When I was searching for a theory of logic I used my teleological method of philosophy, what is the 'purpose' of it? To demonstrate the purpose of logic I shall present a brief summary of my philosophy from first conceptions and determine the exact location at which logic arises (though in truth all of my writing rests upon this logic, a detail which I shall address).

We have a 'tree' like structure of the singularity-form hierarchy which we use to perceive reality. The differences between particular forms by the mind are illusory, they are parts of the same whole. This knowledge of reality I have argued is pre-existent in the mind , what we call a priori. The a priori principle is the Will, the seed as the soul from which reality flowers, rested upon the fundamental ontological principle. The names we give to singularities and forms comes a posteriori, from experience. To elucidate upon this matter it is helpful to set the context to forms of the same singularity, let us say different colours. All colours are one in their concept, but as forms we name them; this naming is not innate but comes from our experience in this world. We can then can speak of the singularity of colour itself, a form in its higher singularity which in turn has also been named. Thus all aside from absolute oneness is named and perceived a posteriori.

It is from this naming that we can isolate a conception of logic. A system of logic arises out of the conception of 'concepts'.
A Concept is a name we give to a necessarily abstracted form, by necessarily abstracted I mean that the form is understood within the context of its singularity which has been conceived.

Again there is a hint of ontology in this matter, the concept being conceived is analogous to the form being conceived, the distinction between concept and form is that a 'concept' specifically refers to a human cognitive perception, I do not attempt to hold a concept as anything higher than its name. The purpose of logic can hence be answered, it is to equate the name with the knowable idea which exists as nameless, pure awareness, its singularity intuited in the mind.

I will take an aside here to answer a query the reader will undoubtedly have. Are not the ideas of 'singularities' and 'forms' also names? I shall state here that the 'singularity-form' system arises out of the conception of 'concepts of structure'. They are reflections of the higher seeds of awareness, of the innate being of the Will.

On Ontological Logic

The conception of a form produces singularity, the conception of a concept produces its meaning. I have discussed before that the meaning the mind associates with a particular substance, its teleology, is its existent reality in itself. All reality is awareness. As such, the conception upon concept and of form both indeed lead to their respective singularity.

To answer how it is that a mere name can gravitate an attached existence other than itself, we will equivalently show that the concepts of 'singularity' and 'form' themselves indeed refer to a higher archetypal reality. This is because their archetypal status ensures that all human perception is shaped through their lenses of awareness, all concepts refer to a form of a singularity which can be considered a singularity in itself.

Recall the Fundamental Equivalence:
The conception of a perception of singularity equates to the existence of the higher singularity. 
Singularity is the absolute totality of its form, the abstracted and generalised state of being. For what I said to truly be meaningful, the concept of singularity must itself really be meaningful which would require my faculty of 'conception' to be a real existent thing in the capacity that we have implicitly assumed.
Now recall the Fundamental Ontology:
We can conceive of conception.
We can now see how the two meta-theorems perfectly complement each other to establish the true nature of ontology, being in itself; a phrase I have used many a time, but now I hope that the reader will be beginning to intuit the precise nature of its reality. I wholeheartedly encourage him to independently pursue this cause to arrive at his own perception of truth.
The cautious reader may again believe me to have pulled a fabric over his eyes and performed a deceit before him, surely I am going around in circles speaking of the same thing and not really transcending it? Let us examine the nature of this claim, to say that conception of conception does not produce a higher reality would be to say that one disagrees with the precise reality of conception that I have described. Perhaps, conception is not truly the leading to totality and that the abstract totality I speak of is illusory; perhaps conception is no more than an identity operator which leads the mind back to the initial substance being conceived upon.
I would not disagree.
Conception could be called an identity operator, singularity could be called the empty set. But that is because we have thus far misunderstood the true nature of what these terms mean; what does it truly mean to say that something is empty? I believe that even the mathematicians amongst us have been misleading themselves, I would say unknowingly but instead I must say rather foolishly. For the empty set is actually everything other than empty - a fact which a learned mathematician would himself proclaim to. If we are to compare my work to this emptiness, then I beckon the reader to compare my work to the ordinal hierarchy. As the empty set has led the mind from the mildest shore of the plain finite, to the distant horizon of incomprehensibly large structures, so too will conception lead the mind from ignoble darkness into the light of ontology. I will elucidate upon this matter in my article 'Discourse on the Empty Set'.

When a concept is applied to itself produces itself again, we say that we have completed an Ontological Circle. An example is awareness of awareness, which is no different than awareness itself. We can in fact produce an infinite sequence of awareness, this sequence in its totality can again be conceived. The conception of the sequence as a whole ties it into its higher singularity, and in turn guarantees its existence in a higher reality. The Ontological Circle is really no more than a summary of the unison of the two meta-theorems described above, of the meaningful conception guaranteed by the Fundamental Ontology to the higher existence guaranteed by the Fundamental Equivalence.

I should address the concepts which do not 'complete the circle', such concepts refer to substances whose existence is dependent upon another singularity conceivable by the human mind. Perhaps it is easier to say that the concepts which do 'complete the circle' as it is, are those which constitute the faculties of reason and logic, I shall touch upon this in more detail in my next article.

There are multiple Ontological Circles which can be completed summarised as follows, in order of their construction:
First Circle - Awareness
Second Circle - Singularity
Third Circle - Conception
Fourth Circle - Ontological Logic
We can  therefore speak of any of these concepts meaningfully for by virtue of there being ontologically complete they are what we can call 'self-transcending', by this I specifically mean to say that the concepts in themselves hold more being than mere words can describe. They are beyond perceptive judgements, for they embody the innate judging faculties themselves.

The fourth circle remains to be completed. My system of logic produces an archetype of understanding through concepts of intuition, it then proceeds to critique this very archetype within the system itself. The system is of the system, again we can produce an infinite sequence of systems studying the preceding structure. In completing this circle we arrive to a logic which can be said to be more than a mere system for it can be said to study whatever system itself. This in turn answers a problem mathematical logic, that each such system must be studied by another system to make sense within its logical universe of construction. In truth mathematicians are basing their logic on naive set theory, the purest mathematical embodiment of abstract intuition. In basing my philosophy solely upon this intuition itself through ontology I have sidestepped the issue entirely, and allowed us to glimpse into the heart of the human understanding - the very fabric of its being rather than its appearance or representation perceived.  

It is here that I should address that my system of Ontological Logic pertains only to the conceivable thoughts of the human mind; at its highest level the system rests upon the throne of being, ontology describing the conceiving faculty itself. It is in this highest concept that I would insist that logically speaking we attempt not to move beyond the concept of the conceiving faculty, for our limited thoughts would do injustice to this seat of God's creation. In fact even in my present discourse I would not claim that the mere words on this paper should gravitate some transcendental meaning, what I truly mean is that the reader through his meditations and contemplations may intuit and realise some greater and profound truth that my philosophy should encourage him towards, this is similar to what I spoke of before that one does not simple know truth, he becomes truth.

It is only appropriate now that we end this treatise on logic with a meta-analysis of my philosophy, as being a concept of truth. What we seek fundamentally is a direct perception of truth, for absolute truth in itself (the conception of truth) is of God's being. This perception exists more so than an idea in the mind as we just addressed, which is why I now present my philosophy as being the logical representation of this perception within the thinking faculties.

Metaphysics of the Will

I could not resist the deep irksome feeling, troubled by a lack of cohesive unity in my discourse on God and the nature of the Self. While the underlying fundamental conception has been presented in the written work, I fear that the reader will not be able to ascertain the true nature of my work, and indeed even I believe that a fundamental aspect has simply remained unwritten within the context of the articles. I seek to rectify this problem by attacking its root, an insufficient doctrine on the nature of the self - most specifically on the Will. The nature of the physical body is not of precise concern to this philosophy, it can be described by a logical system but it is not in my interest to specifically detail it for it should provide no metaphysical value. Simply acknowledging of its existence seems to be adequate at this moment in time. Moreover the mind, in its glorious splendor and horrid complexity can, and will, be tackled in more detail; but it only entails the dynamics of human perception and its cognition through logic within the vast meta-structure of existence. It is this meta-structure, the soul, the Will, that forms the root of existence, and in relating its metaphysical and epistemological being I can bring oneness to my philosophy.

My whole philosophy rests upon a notion of oneness, the conception of 'Singularity' brings it into play in detailing the very being of reality. But how does a conception acquire actual being more than its mere name? The metaphysical grounds of being rests upon a notion of 'isness', that an object simply 'is'. For a singularity to simply be there must be something more than its name, in my previous article 'Metaphysics of Awareness' I justify my belief in all reality being awareness. Thus the root of being for objects is truly a manifestation of the root of being of the self. We ask the question on a higher level, what is it that causes the self to be? What lies beyond the conception of the Soul as totality and oneness of the self, to a very real applied instance of this as awareness?

It is the Will that locks singularity into itself, by this I mean it brings the word into the dimension of its meaning. What is being stated here is more profound than it may seem, perhaps at the onset of my philosophy I believed that the being of substance is none other than the conception in the mind. But really there are two distinct concepts here which must be unified, one is the perception by the mind and the other is its actual being. The two are clearly inextricably linked in that the perception and actual being are both manifestations of awareness, and here it is the Will which precisely represents that paradigm. The will is a metaphysical construction which aids the problem that in existence being spoken of, it is demoted to a property applied rather than an essence pure. What is produced is an underlying field of existence, which we want to hold as innate. In binding this field of existence to the substance itself this innate ontology is manifested, the substance rests upon its own existence which in turn rests upon the existence of higher structures eventually leading to the being of the self.

I now seek to define the Will more abstractly. Previously, we had stated that the human Will is cognition of the self, by which I mean conception of the totality of the self indexed by the self. Perhaps this is a convoluted description of a very fundamental concept; that the self is through the self; otherwise understood as awareness being manifest through ontology. A general conception of the will must be independent of awareness such that it may apply to non-human substances. Singularities in themselves can be said to have an existence rested upon the foundation of their form, the precise idea that we sought; I refer to this as being 'in itself'. Henceforth my definition:
The Will of a singularity refers to its being in itself.
This is not to introduce an external concept of being independent of awareness, which would contradict the very foundation of my philosophy. This is to treat the human Will as the fabric of existence, the Will that any singularity takes to breath life into its concept is a form of the human Will, which could again be said to be a form of the divine Will.

Leading on from the 'in itself' notion, I would also like to review my account of the Fundamental Ontology. I believe its previous expression did not capture the innate essence of being in the capacity that I wished it to.
Previous I defined the Fundamental Ontology as a theorem stating the ability to conceive of singularity itself. Now,
Fundamental Ontology: We can conceive of conception.
Proof: To conceive of conception is to say that there is a singularity of conception. The forms of this singularity would be every possible instance of conception, which would require us to speak of every possible form conceivable. The absolute totality of form conceivable to the mind culminates in the self, since every form in question innately contains the 'aspect of conception' we can henceforth produce a totality of every such conception. This produces the singularity of conception itself, hereby proving the theorem.
This innate essence of ontology is embodied more clearly, in truth that we can conceive of conception should be a triviality to the mind in our being aware of it. The innate 'in itself' archetype is represented in this theorem for we wanted to conceive on every form conceivable, but in the form being conceivable in itself the required totality comes about easily. This is no trick in which I attempt to deceive the reader, the heart of this theorem is indeed subtle; I would beckon the reader to meditate over what existence 'in itself' truly means in order to uncover the meaning.

Metaphysics of Awareness

I would like to revisit the metaphysical notion that all reality exists as awareness. I use the word 'awareness' here to refer to a pure knowing or transcendent consciousness independent of the subject, for this awareness has the capacity to know itself. All perceptive boundaries of our awareness are nullified by the pure awareness itself, in becoming aware of said boundaries they are overcome.

Easier than the task of proving that all existence is awareness is to ask of what exists outside of the sphere of awareness. If we visualise it quite concretely as a sphere there is a system equated with the self, and we ask of what exists outside of it. To give the discussion a more concrete grounding we can speak of the exterior universe or cosmos, a prime question of philosophy pertains to the nature of the 'thing in itself', reality beyond the perceptive faculties of the human mind. If any such reality exists then we visualise it outside of our sphere.

The contradiction is immediately recognised in that we become aware of whatever lies outside of the sphere by its very capacity to be named or cognised about; to visualise our awareness as a sphere limited is contradictory in itself for it would be more accurate to name awareness as the very spatial fabric that the sphere rests upon. As we concluded earlier, any presupposed boundary of awareness is transcended by the very nature of awareness as being fundamental and primordial, resting upon no other than itself. That which is unknown becomes known in its 'unknowability', a contradictory remark at first glance but I say to treat the concept of 'unknowability' as very much that, a concept.

Then it comes sense to speak of what we truly mean by 'unknowability', what the concept actually is. I should conclude here that what we call nothing (to generalise upon the notion of the unknown) is truly something, and what truly is nothing is an inconsistent concept and thus has no meaning in being spoken of. It may seem to the reader that I altogether divert attention and use this to justify my premise that all reality is awareness, but now I point out that what we call 'reality' or 'existence' are human concepts, and whatever substance of awareness is being actualised in overcoming boundaries still subsists as a human faculty. That which lies beyond the human spirit, is cognised by the human spirit, so all reality takes shape out of the mold of awareness.

This is to go beyond the notion that reality is simply shaped by the perceptive faculties, it is to say that there are conceiving faculties which more so than give shape to reality seem to be reality itself. This is truly what I mean by 'awareness'. I should take this opportunity to distinguish between awareness and 'consciousness', in modern times the word consciousness has taken scientific connotation so I let it lie within the realm of science to refer to the product of the biological brain. The awareness which I speak of is beyond physicality, beyond causality and time and space, for it is the singularity; the root of such form.

Those familiar with my earlier work would point out an apparent inconsistency, that I speak of the Divinity as being completely and absolute beyond our conception. This I still hold to be true, but it can be known in some way as being related to the human mind. If we speak of awareness as really being a manifestation of human will, then it is human will which is manifestation of the divine will. We cannot peer into the mind of God, but our own mind, that we are well acquainted with. We conceive ourselves as being form to perceive ourselves as being fragments of a singularity, a higher order and truth that we seek.