The soul or mind reaching towards the formless finds itself unable to grasp where nothing bounds it or to take impression where the impressing reality is diffuse; in sheer dread of holding to nothingness, it slips away. The state is painful; often it seeks relief by retreating from all this vagueness to the region of sense, there to rest as on solid ground.
– Plotinus, Enneads VI.9.3
The Recursive Structure
To know an object in any way conceivable, one ought to think of it since knowledge fundamentally is a cognitive act, and thinking of it as such, one is led to further think about this thought, taking it thus as an object of thinking itself into a new order of thinking by the process which we call abstraction, and as we as thinkers ascend through the orders of thought–this ascending being a function of the mind-thinking–we come to know in a deeper manner what the object of thought is on account of the elaboration of implicit cognitive content, and regardless of the accuracy of this knowledge, it is still considered knowledge that is gradual according to the capacity of our minds to iterate, and as much as it is possible, to extract (in the sense of extrahere) content based on the elaboration within such iterations. As such, there is no binary knowledge and no knowledge that is neutral of mind itself or knowing-subjects, as to say that there is no brute knowledge or knowledge abstracted from all relations to the subject acquiring it or to the object of knowledge, as it is a result of the act of knowing by the knower of the object known, and that activity is thus based on a relationship of apprehension between us as the knowers to the knowledge and subsequently the objects of knowledge. Often many conceive of knowledge per se, but it is not possible to reify knowledge in this manner, since such knowledge does not exist without a mind to know it; of course we are not denying that reality and truth exists independently of us or any rational agents for that matter, but we necessarily interact with them through the medium of the mind and through the act of thinking in order to acquire their knowledge; and thus relationality is constitutive of knowledge-as-acquired.
Though we must ask how do we transition from the 0th level—the level of the object itself—to the 1st level of thinking in which we are taking the initial object into thought, we must first make clear that by object we do not only refer to sensible objects but also to intelligible ones, although still how can the object of sense become an object of thought, clearly then the distinction between the two is not as rigid as it is thought. As far as we have elaborated then, it suffices to say that sensible objects do have intelligible content to be extracted, since either of the alternatives do not work, since it is either that we reconstruct intelligibly that which is sensible and therefore all of our knowledge is synthetic, or that we cannot know sensible objects at all since they cannot be intelligible. Though this extraction involves no transformation whereby the sensible object itself becomes intelligible by ceasing to be sensible, as if thought requires objects to be converted from material to mental, since we have posited that the sensible contains the intelligible and otherwise it can neither be thought nor exist at all, since Being as such is not a sensible object but an intelligible category, and thus in the same instance that we receive the object through senses, we also receive its form and intelligible content through thought, and in this manner we also used the term “extract” since out (ex-) of the sensible we draw (trahere) the intelligible mentally, though the received content through this first act of intellection remains implicit and in need of unpacking and thus the necessity of the aforementioned self-referential unpacking emerges.
We must therefore come back later to the point of the possibility of knowledge as conceived of as being purely objective and independent of whether any minds maintain it as knowledge and taken itself in isolation, but for now we need only reflect on the structure itself, which is unbounded insofar as we can always ascend or descend from the initial thought to the initial object or making that initial thought an object of thought itself as well, terminating the chain at only two possible points: either tautology or contradiction. Or failing to recognize these termini through confusion or incompetence or other particular factors, we can get tangled up in the levels of thought, never finding the stopping point; this termination either tautological or contradictory is due to the impossibility of abstracting a tautology or a contradiction further in a way that can raise us into a new layer of meta-thought—these constitute aporiai, that if unrecognized as such can be the cause of being tangled up in thought unable to terminate.
The recursive structure which we started with can be depicted as such then:
Level 0: The Object (X)
Level 1: Thinking about X = T(X)
Level 2: Thinking about [thinking about X] = T(T(X)) = T²(X)
Level 3: Thinking about [thinking about [thinking about X]] = T³(X)
...
Level n: Tⁿ(X)
Terminus occurs when:
- Tⁿ(X) ≡ Tⁿ⁺¹(X) (tautological collapse: further iteration adds nothing)
- OR Tⁿ(X) ⊥ (contradictory: further iteration yields incoherence)
Where:
T: Ω → Ω represents the operation of making something an object of thought
Ω is the domain of thinkable contents
≡ represents semantic equivalence
⊥ represents contradiction/impossibility
In a way philosophy as an intellectual practice is a prime example of a commentary of and on this same structure, thinking of a whole chain of thought on a particular chain; since one can only think as a particular, and thus through self-referential ascension and descension can trace out the aforementioned thought-chain particular to him, as an instance we can think of the question of Ontology, we elaborate on Being as a whole as a being, or more faithfully to our example of the whole of philosophy, we can think of and elaborate on philosophy itself using philosophy and its history and methods themselves, which would constitute a recursive function of the whole in the whole or the whole in the particular.
Since the philosophers comment and philosophize, which initiates a chain of thinking-hierarchy—the active elaboration by the philosopher—operating on thought-hierarchy—what has already been philosophized and now serves as object of commentary—which open themselves up—referring to the initiated thought-hierarchy—to new commentary, in a way then there will always be new philosophical systems and new questions raised up by such systems that will always need new answers and critiques to those answers, and so on.
And it is quite tempting to look back with anachronistic goggles on the ones who came before and see what they did as trivial when we are priming ourselves for the same judgement in the future by attempting to appropriate a God’s eye perspective or by divorcing early thought from their intellectual hierarchies—not in the sense of institutions but in the sense of abstractions and concretizations—but each doing so from his own chain of commentary that is contained within a specific mind which would like to see progress as inevitable, so this supposition is what is read back into the past, when if we brought the ancients into the present they would see a sort of philosophical decay—as far as they are concerned—in what we have done thus far and could only describe it as the same sophistry which they—Plato and Aristotle especially—have dealt with—and with extraordinary insight; this is because one unknowingly deals with the whole when thinking, thus he also deals with thoughts he has not even thought of.
Tautology, Contradiction, and the Geometry of Mind
Now aside from this brief note on philosophy, we must also clarify what is thinkable exactly, by which we mean what makes an object thinkable at all, and what makes the unthinkable unthinkable—yet we can “think” in some way of the unthinkable, since merely by invoking it we are thinking of what it is, or rather what it is not. One clear boundary between the thinkable and the unthinkable can be drawn through Being, since that which exists is thinkable, yet we can also think conceptually of what does not exist. Here we must distinguish what is conceivable and inconceivable within the non-existent, and even within the inconceivable there is variety, since that which is logically impossible on account of essential contradictions in its formulation is not the same as that which is irreducible to thought due to transcendence or abstractness, such as Being itself, which cannot be a being as such but rather is the ground of beings which grants them so to speak the quality of existence; so then we cannot truly conceive of Being as such but all that we can know of Being is inferred in some way from beings which draw their quality from It. And in this way what is thinkable is that which encompasses what is and what is not though only what is conceivable in that which is not, and even then the unthinkable which encompasses the non-existent–both logically impossible and the irreducible–can be thought though indirectly either by negation in both cases or by analogy from the immanent to the transcendent which is not reducible.
Moreover, when it comes to this self-referentiality we have elaborated on, knowledge is often defined, by those who think of it as independent of states and capacities of mind and thinking agent and abstracted from relationality to the object and subject of knowledge and knowledge itself, so strictly as to be impossible to grasp, and to make the word ‘knowledge’ itself superfluous since it would refer to nothing conceivable by any faculty, since most if not all of what we can claim as knowledge cannot reach the standard that is set as such, when knowledge as we defined earlier is much more attainable; and also it is not a dichotomy of either knowledge or ignorance but a matter of degrees, since it depends on our mental explorations of the levels of self-referential thought that resolves into and terminates in generative tautologies—that is, tautologies that generate insight about the object of knowledge, and the structure of said object, and also of the whole structure of the world on account of them being pregnant with content, as any knowledge of a thing is in a way a knowledge of the all, given that all things are interrelated in a web with each node being wholly impossible to isolate in itself. This generation of insight through the terminus of tautology is done by revisiting lower levels with the knowledge acquired through the meta-levels, thus transforming our initial considerations like acquiring a microscope made from finer materials in order to penetrate through sight the microscopic objects that were previously invisible.
These generative tautologies that we mentioned are very prevalent in the foundations of philosophical thought, and of course we are not using tautology in its narrow logical sense but in a broader and structural sense; we might illustrate it using examples to show what we are talking about: Parmenides’ ontological principle establishing the notion of Being as such “What is, is”, the law of identity in logic, “A=A” which is structurally—and we might say even in terms of content—the same as Parmenides’ starting point, though in the context of logic, Descartes’ “Cogito ergo sum (I think therefore I am)” establishing the inseparability of thought and being, Aquinas’ “Deus sit ipsum esse subsistens (God is subsisting being itself)” which establishes what is proper for the first principle, and so on. On each of these examples great edifices of philosophical thinking are built and maintained even if on first glance they seem to be terribly trivial and uninformative, so as to be functionally meaningless, and this might be responsible for the dismissal of philosophy as word-games and sophistry by most of those uninformed about it; even what I am doing here might seem so to untrained eyes as well.
Now for each of these examples, and what makes them truly generative, is that they unfold into a whole network that is an image of the whole world on account of them being maximally abstracted and on account of their relationality to the whole; for instance Parmenides’ principle unfolds the whole of Ontology from then on as well as the law of identity for logic, and Descartes’ cogito for the whole tradition of rationalism, and Classical Theism with regards to Actus Purus as demonstrated by Aquinas—though the same idea is present before him as well. This seems to be in alignment with the self-referentiality of thought, the result of deeply hierarchical chains of thought that terminate in tautology that contain within them the seed for the whole chain, as opposed to empty tautologies or ones that are definitional that operate on first-order or second-order self-referential levels mostly that either terminate on those same levels or can unpack one or few layers of meaning from them, on account of their maximal particularity and isolation from the whole.
Now when it comes to our examples, it is natural to ask about the question of incommensurability, and that these can be doubted and have been doubted, and even convincingly argued against. But it is worth noting that often the disagreements is not due to the form of them as such or their structure as such, but rather to how they are understood by the ones who posited them and the kinds of conclusions that these same ones thought that they thought followed from such structurally tautological statements which are surely open to interpretation and to disagreement.
These two, compression and unpacking, are both analogous to the ascent and descent mentioned earlier, both are the same as abstraction and concretization of ideas resulting from thought, and both necessary for deeper knowledge of the world and of its beings, parts and aspects which pull the mind constantly into the act of contemplation, just as inhaling and exhaling are both necessary for breathing in order to live, something that is lost on both Idealist and Empiricist tendencies which content themselves with unidirectional movements; since just as it is necessary to arrive at the terminus by abstraction, it is also necessary to descend from the summit with the new insight found therein.
On the other hand, our ascent through the levels of self-referential thought can terminate at contradiction, which is opposite to tautology, and as such also shares the same properties as it, in an inverted sense nevertheless; and thus if a tautology contains the seed of the whole network and structure of reality, contradictions serve the same purpose but for the boundaries of the world generated and what lies beyond the boundary of the world, that is non-being. This is because a contradiction shows not where knowledge can be but where it cannot—for example, to assert a contradiction as knowledge is not to claim knowledge but its lack, but not in the negative sense, rather that no knowledge can apply to it, as in the case of a “square circle” which marks where geometry cannot be geometry, since the starting categories are geometrical though the resulting synthesis is not a conceivable category at all, thus not geometrical in any sense. And thus the inversion of the sense of the properties of the tautology is manifest in these dialectical opposites: boundary as opposed to foundation, non-being as opposed to being, outward as opposed to inward, generative of limits as opposed to generative of horizons.
This can be thought of as generative either inwardly in the case of tautology or outwardly in the case of contradiction as we have said, and the same analogy of inhaling and exhaling operates not only when it comes to ascent and descent but also on the level of tautology and contradiction; the two levels being axes of the mental space, horizontal for tautology and contradiction, vertical for ascent and descent, at the center of which lies the descended transcendent which is the concretized abstract and the reconciled contradiction, that is the contradiction integrated and resolved where all of the four are unified and made truly immanent and accessible.
ASCENT
(Abstraction)
^
|
|
CONTRADICTION <────────●────────> TAUTOLOGY
(Boundaries) CENTER (Foundations)
Non-being | Being
Outward | Inward
v
DESCENT
(Concretization)
Now by the nature of our thinking and through expanding on this, not that we have acquired a view from nowhere but within the chain elaborating on it itself, this itself is not only the superstructure of the whole chain or rather cruciform hierarchy, but is made possible also due to this structure operating on each level and manifesting itself in microcosm and macrocosm of self-referential thought; each act of meta-thought is itself thought not divorced from thinking as such but doing an act of the same nature but of a different mode. To claim otherwise is to make the possibility of thought given what we have already elaborated incoherent and access to other meta-levels impossible.
On Objectivity and Degrees of Certainty
Furthermore, considering this account of thought and knowledge, what would these things mean for objectivity of knowledge, absolute knowledge and certainty, a question which we skipped earlier? Now as for objectivity, it is that when a statement is true to all minds, not that one disagreeing makes it false, but rather even it is true in spite of disagreement because it is necessary for any process of thought and if negated makes rational thinking impossible, and even one who disagrees implicitly affirms them and uses them to make the argument for their denial; and usually what is termed objective truth is a generative tautology or is derived directly from it or from the structure it generates, and we are certain insofar as we know these, but the further we descend the less the degrees thereof on account of the multiplication of interpretive choices, contingent factors and other variations. And in this sense objectivity and certainty also come in degrees depending on how the object of these predications relates to the whole, in the sense that what is subjective only relates to parts and particulars and thus is less certain as opposed to that which relates to the whole and thus is more certain and thus what is objective is not what does not relate to anything but rather what relates to the whole in its fullness.
Let S = "Rational thought requires principle P"
If someone denies S, they must use rational thought to deny it
Rational thought requires P (by S)
Therefore, denying S requires using P
∴ Denying S presupposes S (performative contradiction)
However one could ask how come, with all the various generative tautologies we can formulate, one can ground them? Certainly it is not that each one stands by itself, for then we would have uncontrolled pluralism opening the floodgates for ‘anything goes’ relativism. There must be an underlying principle that unites all such tautologies.
Thus from what we have gathered and the definitions and elaborations we have provided thus far with regards to the structure of thought and its relationality and self-referentiality, we would say that all tautologies and contradictions stand on a unified ground. Tautologies express the indefinite content of the inward whole—indefinite in the sense that it encompasses all which cannot be distinguished within itself or into aspects of itself. Contradictions express the definite boundary distinguishing what is outward from what is inward. Together they make the indefinite definite without emptying it, rendering the content intelligible and unifying the many into one, through integrating both aspects: what being is and what it is not.
This unity is fundamental in the sense that if it is not acknowledged, then the edifice of the structure collapses unto itself, in the same way that a mountain necessarily has a highest summit, not that other summits might not exist but that highest is necessary to recognize the other summits as such, to relate to our structure then, both tautologies and contradictions might not be recognized as relative termini without an Absolute Terminus that allows the aforementioned termini to be properly termini in the relative sense. This is because in the same way as the notion of height presupposes the ground level by which it is recognized as height relative to the ground, plurality of nodes of cognitive hierarchy presupposes a unity at the center by which these nodes are considered local centers receiving their centeredness from the unity. And in these images we recognize the unity as both Absolute Ground and Absolute Summit on grounds that it is that to which summits and grounds are relative, and neither summit nor ground on grounds that it is unlike the relative summits and grounds.
This one—the synthesis of both being and non-being yet neither, and both thinkable and unthinkable yet neither, as it transcends them—is the ground on which all tautologies and contradictions stand. They are expressions of the one: each tautology expressing it in different modes and aspects (what the one is, under different determinations), each contradiction expressing what the one is not (the boundaries beyond which the one does not extend). Thus from one transcendent ground arise multiple generative tautologies, not as arbitrary plurality but as the one expressing itself in different aspects, each capturing one facet of what cannot be fully comprehended in any single determination.
From this grounding we can see that these multiple tautologies operate within one unified framework and are not incommensurable, and are thus synthesized in one reality properly called, where the plurality serves the unity as different modes and aspects of the one transcendent ground, each capturing what cannot be fully comprehended in any single determination.
Modal Ontology of Self-Knowledge
But one must ask as well about the subject doing the act of intellection, as any act of thinking is proper to a mind which always subsists in and is proper to a rational agent; this thinking agent can think about other agents as objects of thought or of himself, this thought of one’s self is itself self-reflexive and can also be self-referential in deeper self-reflection. It poses a different kind of structure and one that is deeply more layered than the one we posited before, since now the subject is both thinker and thought at the same time, both the subject and the object, and definitely the modes of self-referentiality multiply greatly when we consider the subject operating not just as thinker but also as an emotional, willing and acting agent, but for the moment we only will consider the mode of thinking.
This complexity I mentioned as opposed to thinking of external objects is due to the fact that in object-oriented thinking the subject and object are clearly distinct and separate, but this is not so for self-reflexive thought since the knower and the known are one and the same; while of course it is practically possible, since we operate in this mode of thinking on a daily basis, but we must examine how it is possible for this to be.
It is that subject and object are not ontological categories but rather modes of being, by which I mean that they are not fixed instances of Being (e.g Man) but rather flexible functions (e.g Man-as-teacher), because if it were an ontological category, it would be the case that thinking of one’s self divides the person irrevocably into two and with each level of thought it further splits into powers of two, which would be very much absurd, and it would also be the case that a being or a person can exist in different modes simultaneously and also that in each mode one subsists fully for it to be possible to know one’s self.
- Thinking requires distinct subject and object
- If subject/object are ontological categories:
Level 1: Person splits into 2 entities (subject₁, object₁)
Level 2: Each splits again → 2² = 4 entities
Level 3: Each splits again → 2³ = 8 entities
Level n: 2ⁿ entities
- But: Person remains numerically one
∴ Subject/object cannot be ontological categories
This also clarifies that we think even for the case of objects distinct from our person in their modality and not on the level of their ontology; this is why it is possible to maintain that reality exists independently from mind while also being accessible through the mind without any issue of an unbridgeable metaphysical gap. This is because the ontology itself is not an object-to-be-thought, since for an object to be thought coherently it must first possess an ontology, and not that thinking of it confirms its being but rather its being makes it thinkable, therefore through thinking of the modes of ontology we come to know ontology, but not in the absolute sense or as something that follow directly but only by analogy.
But for the case of self-reflexive thinking itself, what happens is that when one ascends through meta-levels thoughts of himself, he introduces modalities in which he is fully represented and thinks and is thought about within those, without requiring to step outside of one’s self and see himself from a third-person perspective.
Thus this also sheds light on the ontology and modality of objects both of sensible and intelligible natures, which both exist independently, and are distinct in nature but unified through the mind, which thinks of their mode of being in the same way, otherwise the sensible must be ontologically transformed into the intelligible to be thought, and it is not that the ontology of the objects is unknowable, but that the ontology of them per se is not an object of knowledge in the first place, but their mode of being is the object of thought and knowledge, and as such, it is a category error to talk of knowing the ontology itself instead of the mode; it is as if one talks of hearing colors or seeing sounds.
And as far as we have gone, to think of these things I’ve written of is to demonstrate what is thus written—not that it is complete or the terms used are set in stone, but the experience is the same whether it is expressed the same or different. Or in other words to think properly is to know the hierarchy of thought, and to be, in the proper sense is to know the hierarchy of being.
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