544  According to the embryogenesis of the concept,i* differentiation manifests necessarily at first as polarity, and the interface is the cultivated/cultivating third in the recognition of the unity and triunity within the terminal and irresolvable ends of any duality.  Recall also that some of these root-level, or meta-paradigmatic polarities are operationalized in the Vision-Logic Coordinate System, with its transitive, immanent/transcendent omni-, and uni- axes—and their respective triune interfaces, such as the transitive-plane and the platonic sphere, for the omni- and uni-axes, respectively.  As we have just seen in Spinoza’s Attribute Polarity, the omni-, and uni- interfaces each play critical roles as the cultivated thirds at the crossroads of the ontic/epistemic and subject/object polarities.  And, as we have also seen, the interface of the immanent/transcendent uni-axis (the spheroidal and bounded “finite unity” and “unit-identity”) is the very beginning of Mathematics in its holarchical embryogenesis as found in Interface Mathematics, just as it is the beginning of representation or mimetic consciousness (roughly the epistemic) itself in the nucleation of observability.  Indeed, Interface Philosophy itself is a “vision-logic interface” for transferring the framework of a new, nondual way of rational, trans-rational and even empirical thinking, into various realms of application.

545  Interface Epistemology, therefore, explores the relative interface and relation between the two fully relative “worlds” of form—the epistemic and the ontic; representation and sub-representation; the map and the territory—but it does so in the context of the Univocity Framework which embodies the essence of the principle of nonduality in the polarity of polarity,i* keeping open and ineffable the absolute scope, while exploring the relative world of real form (e.g. Brahma) and the endless possibilities of its real representations and illusions (e.g. Maya).

549  The task of post-Kantian, post-ontotomy Interface Philosophy, then, has been to bring the ontic back out of the absolute scope, and to provide an epistemic interface for exploring this lost continent.  And in this process we will continue to demonstrate that the ontic is not the ineffable, untouchable Absolute, but fully within the relative and effable world of form embraced in the very interface of Maya and Brahma.  And indeed, this interface of Maya coming to know Brahma grows ever larger as more and more unknown territory is uncovered daily and become known.  But more to the point, in our relative freedom to explore the lost ontic continent of Brahma in epistemic union with Maya, we will then be equipped to explore the lofty heights to which only Maya can lift us.

557  As the attentive reader can see, in our EOTC of the epistemic categories, we now have a fourth and even a fifth level differentiated-and-integrated into our original triune holarchy; the epistemic subject, the subject/object interface and the epistemic object.   So now we have the fivefold delineation of epistemology; (1) the ontic, (2) the ontic/epistemic interface gradient, (3) meaning, (4) the meaning/language interface gradient, and (5) language.

558  The Absolute scope breaks into the relative scope and then into the ontic/epistemic (Brahma/Maya) holarchy/polarity.  From there we expand the category of the ontic/epistemic interface gradient and break it into the intra-objective/intra-subjective sensory-mnemonic primitives, and complexes. Via Aristotle, we have also expanded the category of the epistemic first into meaning and then into higher level language and into the inter-objective/inter-subjective meme-complexes, such as schools of thought, where language and meaning unfold each other into ever higher and higher levels of complexity.  The Absolute scope breaks into the relative scope and then into the ontic/epistemic (Brahma/Maya) holarchy/polarity.  From there we expand the category of the ontic/epistemic interface gradient and break it into the intra-objective/intra-subjective sensory-mnemonic primitives, and complexes. Via Aristotle, we have also expanded the category of the epistemic first into meaning and then into higher level language and into the inter-objective/inter-subjective meme-complexes, such as schools of thought, where language and meaning unfold each other into ever higher and higher levels of complexity.

563  The Absolute scope breaks into the relative scope and then into the ontic/epistemic (Brahma/Maya) holarchy/polarity.  From there we expand the category of the ontic/epistemic interface gradient and break it into the intra-objective/intra-subjective sensory-mnemonic primitives, and complexes. Via Aristotle, we have also expanded the category of the epistemic first into meaning and then into higher level language and into the inter-objective/inter-subjective meme-complexes, such as schools of thought, where language and meaning unfold each other into ever higher and higher levels of complexity.

567-568  So, in the integration of AQAL metatheory with Interface Philosophy, we can say that any epistemic mapping of an Immanent/Transcendent axis and coordinate into a transitive-plane of relation must include the four quadrants if it hopes to be all-inclusive and “integrally informed” of the fundamental perspectives of human (and perhaps indeed nonhuman) awareness.  To be sure, we can simply imagine that in the exfoliation of any transitive-plane from any Immanent/Transcendent axis, the AQAL map is implicit, and can be made explicit in order that one quadrant not be neglected or reduced out of the picture for the sake of any of the others.

571  So, the danger with these statements which seem to imply the absolute fundamentality of epistemic points of view or perspectives without making clear the acknowledgment of the ontic-level embodiment in holons and bounded form—and indeed the interface and holonic gradient of emergence necessary for representational embodiment itself—is that IMP can seem to collapse to an epistemic absolutism.  The hope, therefore, is that IMP will clarify this ambiguity in the explication of the ontic-epistemic polarity, in its exploration of “post-ontology” with the corollary that “All is holonic.”

583  Indeed, the most consistent way to understand this periodic quantization of holarchical involution and evolution is that the enfoldment of infinite holarchic complexity itself maintains the extreme unit integrity of the simplest holons or units, as the energies maximally emergent at the focal point of the unit become internally refracted and reflected toward harmonic self-reinforcement and self-stabilization (e.g. inertia).i   In this sense we have another true polarity: Holons of MINIMAL outward or inter-unit complexity occur precisely at the focal point of MAXIMAL complexity, where they can finally harness the complexity of the infinite holarchy into their own self-stabilization and inertial integrity.  Simplest bodies, then, are simplest precisely because they are internally the most complex and self-organized.  In this sense, evolution itself leads directly to involution at the next level.  Indeed, evolution and involution are the two sides of the same coin, because it is increasing complexity itself— evolution—that allows involution to proceed toward the self-centering, self-harmonizing, self-focusing of its ever higher ground-levels of simplest bodies.  Evolution, then, is involution seen in reverse.  And we can see this as a harmonic corollary to our Principle of Immanence in Transcendence.  Just as immanence is transcendence if seen from “beneath,” so too involution is evolution if seen from within its unfolding process.

586  So the meditator is approaching the “ground of being” in many senses.  He is approaching the ground of holonic simplicity itself (minimal depth), in minimizing the complexity of his thoughts—breakingthem down or “pulverizing” them from mnemonic complexes to primitives—while at the same time approaching a general self-similarity (or roughly analogous “psychic” form) to the actual internal structure of the simplest holons with their maximization of the internally inter-harmonizing patterns of infinite energy at the emergent focal point at the interface(vinculum) of awareness itself, of the implicit singularity of the immanent/transcendent uni-axis.  It is this implicit singularity (or an “infinite number” of them) at the heart of his bounded being—reflecting through the Principle of Absolute Reversal into the transcendent—which he later acknowledges and analogizes as he approaches the immanent limits of his own experiential form and boundary: his sub-mnemonic interface.   A finite unity cannot experience the absolute scope of the infinite because experience itself is relational and relative.  Or, put another way, one must experience the absolute through the relative; one must experience the infinite through the finite, which results in the boundlessness of the indefinite, or in the nondual resting in the infinite and finite as one.

608  As we have seen, it was this imperfect replication itself—this mutability—which was the initial creative, learning and intelligent function of the code.  A perfect linguistic replication can never make mistakes and can never learn.  It can never create or solve problems, but can merely recapitulate itself to eternity.  In this “negative gradient”—from a sterile, immutable Eden into the fecund Earth of continual creation—evolutionary intelligence has “fallen.”   But critically, for our purposes DNA is a causative mnemonics; a living, breathing, replicating language of memory.  This “imperfect” memory allowed the first level of geologically slow creativity and intelligence to emerge.  Sexual reproduction and selection, on the other hand, can’t so easily be seen as merely a failed causative and replicative language, but rather a successful collaboration between two organisms to bring about a new one.  But not only that.  It is a collaboration and a communication between organisms and their code.  An intelligent cultivating third, interfacing between two of its own intelligences, injecting their combined intelligence back into the evolutionary stream of creation which engendered their communicative dance in the first place.

610  Strangely, but as we may by now expect, the new direction in cellular biology directly supports an Interface Philosophy or Epistemology of emergent intelligence, creativity and knowledge.  Cellular biology is indeed moving away from the idea that the cellular nucleus is the driving intelligence or “brain” of the cell, and toward the notion that the intelligence is actually found in the outer cellular membrane itself; the interface where primitive sensation and its encoded reactions (rudimentary genetic-mnemonic primitives) takes place.

616  And so we can see in the very rudiments of cellular evolution the interfacing between “behaviorism” and the “subjectivity” which the extreme versions absolutely ruled out.  Without sensation—which emerges at the higher levels into the more complex forms of representation—there can be no intelligence to effect and inform complex behavior.  And indeed, without the possibility and actuality of early reflex forms of behavior coupled to sensation, there would be no reason and feedback mechanism for sensation to evolve.  As might be quickly intuited, this is a corollary to our symbiogenesis of subject and object.

620  The concept of time, for example, seems fundamentally inherent in human consciousness, but in its embryogenesis there is no point at which we can de-couple the concept of time from the perception of duration, and even deeper into the experience of evolutionary intelligence with the forms of duration.

477   Thus the naked paradox dissolves, as every child knows it must.  Zeno’s immovable plurality of instants is merely a con-fused abstraction.  An unreachable aspect of immanence abstracted from Parmenides’ own Being now.  Parmenides and the child have won—and thanks to the countless man-hours of the myriad regiments of mathematicians, scientists, philosophers and artists, inspired by Zeno himself into working out the details of this con-fusion—tuning and triuning the Zenonian interface itself—we can now say exactly why.

482   …the paradoxes themselves put a pressure on reason to work themselves out into nondual recognitions of the fundamental polarity of conceptualization itself.  With Galileo’s paradox, for example, we saw that the polarity of infinity and number worked itself out through the Cantorian transfinite orders and arithmetic of the infinite, and into the nondual-rational Triune Infinite.  And through Zeno’s paradoxes the mathematics of the continuum worked itself into its own ordering of the immanent infinities or singularities of the irrational and transcendental numbers, and now into philosophical understanding with the triune interface between the two fundamental VL- axes of conceptual thought itself.

486  …we delve into the traditional details of philosophy itself, instead of the areas “outside”, or in the external direction away from philosophy (such as mathematics) into which philosophy often strays—offering its integrating perspective by weaving a tapestry of concepts.  The problems we will encounter—with our nondual-rational integrating perspective and meta-philosophy now intact—are the common ones which, solved long ago in essence (by the nondual philosophies in general, for example), seem to persist in philosophy classes, their dualisms grotesquely exaggerated and caricatured so as to offer eternal resistance and practice for the budding philosophy student—and perhaps to keep the professors and philosophy itself employed in the modern academy.  These problems include—the “mind-body problem”; the problem of free-will and determinism; the nature of knowledge (i.e. epistemology); as well as the problem of nondually integrating and inter-relating the philosophies of Spinoza and Leibniz, the two main nondual rationalists.

493-494    In an attempt to reach what Deleuze would call the “plane of consistency” in Spinoza’s thought, we must give Spinoza the benefit of the doubt and assume that he included both the ontic and epistemic (or “Subjectivist” and “Objectivist”) aspects in the definition of the attributes for the purpose of conceptual utilization and clarification, rather than accidentally or for the purpose of confusion.  Therefore, if we can find a way to make sense of Spinoza’s text as it is, that is to be preferred, even if it requires much conceptual infrastructure or meta-perspective to see it explicitly.  The remainder of this discussion, therefore, will be focused on the ways in which we can understand, both visually and viscerally (intuitively), how the intellect, and indeed all modes at all levels of emergence, must be formed in such a way as to perceive the essence of Substance through these two ways, aspects or attributes.

495 – 496  …this nominal problem is due to the hidden nature of these dual polarities and the subsequent modern tendency (“flat- land materialism”) to collapse the objective into the ontic as foundational and to raise the subjective to the secondary, and at best emergent epistemic.  The other modern tendency (“idealism”) is to react and counteract this object-reductionism with its opposite replacement in a flat-land subject- reductionism.  This gives us the opposing equations, object = ontic vs. subject = ontic, which gives us the relational nuclei of the twin poles of this exegetic rift.   The nondual-rational  solution to this problem of foundational imbalance and tacit-dualismii* will be facilitated, naturally, through use of the VCSiii* and the Univocity Frameworkiv* as a conceptual reference system for understanding and explicitly maintaining the crucial distinction at the crossroads between these con-fused orthogonal polarities—ontic/epistemic and subject/object, as well as the univocal absolute/relative—and for keeping these polarities from collapsing upon themselves, and/or into each other.

498   In Spinoza’s time in general, and in his writings, however, the term ‘objective’ referred to the observations of external reality rather than to that reality itself.  Therefore, ‘objectivity’ referred to our knowledge of the external world and ‘subjectivity’ referred to our knowledge of the internal world.  …With the Newtonian scientific revolution of the 17th century, however, and the wild success and ascendancy of pragmatic mathematics, physical principles and Laws of physics—at the expense of causal or ontological speculation and explanation, and its roots in scholastic terminology—the meaning of ‘objective’ slowly became synonymous with the ontic, and came to refer to the reality itself whose Laws we felt we were uncovering, e.g. Galileo’s book of Nature written in the language of mathematics

499  The new question at the core of the problematic exegetic division then becomes; Are the dual attributes essential to the nature of Beingiii (ontic) or are they aspects of a bifurcation in perception/conception (epistemic) and secondarily imposed on sub-representational reality (the ontic)?

500-501   In concurrence with Gödel’s Incompleteness Theorem and with philosophical “Fallibilism,” the mind-body problem now merely demonstrates the limits (or fallibility) of all representational systems— whether subjective, objective, mathematical, linguistic, aural or visual.  No system of explanation and no method of observation can ever be absolutely complete and infallible because all representational and explanatory systems are based in the epistemic realm as emergent finite abstractions and observations.  In Spinoza’s system, however, the immanent causation of Substance is infinite in extent and resolution (i.e. continuous—via our Principle of Nondual Rationalism, infinite division equals indivisibility) and thus cannot be encapsulated by an abstract and finite representation. In this way, it is only natural that objective accounts can never absolutely explain the subjective experience (and vice versa), and indeed the need for them to do so becomes revealed as a simple categorical error between the objective and subjective modes of knowing.

509-510   …the perceived ambiguity in Spinoza’s definition of an attribute— which appears to suggest both “subjective” and “objective” interpretations simultaneously—is rendered coherent when their dual ontic/epistemic nature is exposed as a necessary formative nucleated polarity inherent in the very nature of active Substance as it is modified (evolved) along a morphological gradient into an increasingly complex series of ever more cognitive/representational modes (brains/minds).  And simultaneously, the false dichotomy of the exegetic rift between the Subjectivists and the Objectivists is rendered visible and defunct when the epistemic and ontic understanding of an attribute is at once made clear.  Without this symbiotic inside/outside distinction, modes simply cannot self-stabilize, exist and be perceived or conceived.  And a modification with an inside and an outside is always extended and thus capable of interrelation with other modes in the field of extension.   …Despite the duality inherent in the formative protocols of nucleation, Substance itself always remains univocal and continuous (though not homogenous) and this continuity (infinite division equals indivisibility) is in fact key to its formative protocols and the emergent interface separating, uniting and enabling the nuclear differentiation in the first place.  …Materialists (or “Subjectivists” in our outmoded Spinozistic nomenclature), though they sometimes find a deep resonance with Spinoza, nevertheless often have difficulty with the idea that Spinoza’s attribute of thought pertains to the ontic level (erroneously assigned to the Objectivist camp in the objective-ontic shift) and thus to all modes, whether or not they possess a brain and thus a perceiving intellect.  To them this seems to mean that objects such as trees and rocks are sitting around thinking, like we do.  However, at about this point Spinoza says, “The propositions we have advanced hitherto have been entirely general, applying not more to men than to other individual things, all of which, though in different degrees, are animated [animata].”  This “anima” is the immanent active causation within all things (corresponding to the existential idea that existence, an active property, precedes individual essenceii, and as Spinoza says in his Short Treatise on God, Man and His Well-Being, “since substance is [the cause] and the origin of all its modes, it may with far greater right be called acting than passive.”

511-512  In other words, the human mind (the “idea of the body” or the idea that the body has), as epistemic and representational, emerges from the body itself as ontic, but both stratified ontic and epistemic aspects of attribute-neutral modality of the body are subject to the vertical bisection of the twin attributes, Thought and Extension, since holonic nucleation, and the polarity of the within and without, occurs at all levels of the body, ontic and epistemic (recall Figure 65, p505).  …The dual (within/without) aspect necessarily inherent in the formative and descriptive (ontic and epistemic) essence of all modification quite simply accounts for the exact “order and connection” that both the “modes of thought” and the “modes of extension” adhere to (see Figure 67, below).  “The order and connection of ideas is the same as the order and connection of things” and simultaneously it explains why the two attributes must function in parallel isolation from one another; this is because as a localization you can’t simultaneously be outside the mode, viewing it under the aspect of Extension, and inside the mode as the intrinsic experience itself.  Or more simply put, you can’t be both outside and inside yourself at the same time.  However, Spinoza also says, “thinking substance and extended substance are one and the same substance, comprehended now under this attribute, now under that. So, too, a mode of extension and the idea of that mode [[a “mode of thought”]] are one and the same thing, expressed in two ways. The “idea” of the mode, as a modification of the attribute of Thought, is this intrinsic existence or the experience of being the mode itself.

514-515 The mistaken reading of the attributes is the assumption that objectivity and subjectivity—the twin aspects of cognition, representation and the epistemic—are both subsumed under the attribute of Thought.  The attribute of Thought, it must be clear, is not the common notion ‘thought.’  Objectivity is the epistemic level of the attribute of Extension and subjectivity is the epistemic level of the attribute of Thought.  The “crossroads” of the two polarities—the attributes and the ontic/epistemic—arises at the emergent monadic interface of every cognitive/representational holon or mode, but the distinction must be kept from collapsing into a con-fusion between the omni- and uni-axes of the ontic-epistemic and the subject-object polarities, respectively.  …This individual interface is the reductive goal of the Cartesian “cogito”; “I think therefore I am.”  But the attempted reduction of relative knowledge to the absolute scope takes us not into the certainty of any of the truths of form outside the self, but to an absolute certainty of only oneself—the solipsistic self-absolute because the only thing the self can know absolutely is the representational interface itself.  All else is projection.   Through the principle of absolute reversal, however, of taking the self- boundary to the absolute, it dissolves its other and fades into the omni-non of nihilipsism (self-annihilation).  If there is no other, there is no self—just as in the omni-local if there is no distinction between one location and all others, there is no location referenced at all.  Ultimately we come to the annihilation of the self-distinction and boundary itself with the elimination of polarity.  This is the “cogito of the dissolved self,” roughly.

526   We have seen that both the descriptive protocols—subjectivity and objectivity—emerge from the morphological gradient of emergent “excellence” (with its polarity of formative protocols) into the representational realm of the epistemic.  Thus both subjectivity and objectivity are finite (or indefinite) systems of relative knowledge about the relative and infinitely complex, intrinsic or extrinsic world of form, respectively.   …In terms of Spinoza’s Triune Infinite, taken as an “organizing principle” behind the Ethics, the attributes are the aspects of The Infinite (Substance), as seen from within or from without any particular modification.  This morphological and polar explanation, then, gives a concrete reason why Spinoza only gave us two of the infinite, or unlimited attributes of God—a problem which has vexed Spinozists from the beginning. …With the concept of “the nucleation of observability” it has been pointed out that all systems of knowledge emerge at the sensory/representational (mnemonic) interface of a cognitive mode, and this emergence is always transcendent to a deeper level of immanence, on the I/T omni- or uni-axis.  We have also seen this as the simple meaning of Deleuze’s “transcendental empiricism” in that perception/conception always begins from a plane of immanence that is already necessarily transcendent or emergent from deeper immanence.  Such an immanent plane, we have also seen, is necessarily, at the same time, a transitive-plane.  Empiricism, therefore, being derived ultimately from emergent/collective perception, is necessarily inherently emergent or transcendental, conceived as a transitive plane on the immanent-transcendent omni-axis, for the collective of empirical data, and a spherical interface on the uni-axis, for the empirical surface of the individual.

528   According to this nondual-rational point of view, then, both the materialists and the idealists (or the Subjectivists and Objectivists) make a common mistake.  They each fail to understand the proper role of the subjective and the objective as fundamental descriptive protocols in the epistemic and relative realm of inherently limited and fallible representational abstraction.  They therefore each take one epistemic protocol or the other and erroneously project it down to the ontic level, toward the foundational absolute scope, in exclusion of the other. This unbalanced view is akin to claiming that outsides can exist without (or prior to) insides or vice versa and that the entire universe is ultimately composed only of either the outsides or the insides of things.

529-530   Both of these views, we can see, ultimately (when taken to their extremes) collapse into the “cogito of the dissolved self,” the solipsistic reduction to the ontic-epistemic boundary of individual experience and the  absolute reversal from this solipsism to the identical opposite omni-non of nihilipsism, when the absolute self dissolves and digests its defining other.

532  In the complexification through evolutionary intelligence of this symbiotic subject/object interface into the more abstract representational forms of awareness and consciousness, more and more layers of abstraction are added to the pre-existing massive parallelism of the subconscious and preconscious homeostatic organism.  Each new layer of abstraction, while more complex than the last, generally serves to consolidate and reflexively integrate the previous, more ‘primitive,’ layer until finally reaching the level of ‘excellence’ of the human being with its cerebral cortex and unitary experience or “stream” of consciousness.   …The human organism has thus emerged, ultimately, at the level of abstraction of the linearity of language and the highly focused ‘inner voice of consciousness’ or ‘the stream of thought.’i  This emergent linguistic linearity enables functionality in the form of complex strings of internal meaning and external communication (both of which, in the same process at this higher level, are also symbiogenetic).  Through a similar symbiogenetic feedback process, external communication and internal abstract conceptualization enables a further degree of rigor and linear complexification in the thoughts themselves (mnemonic complexes) due to the internal and external structure of the language in which the thoughts are functionally ‘encapsulated’ and reflexively conjured.

261 -262  From this pre-differentiated multiple aspect in the ALL we begin the embryogenesis of operation and number.  But only through a “breaking of symmetry”—from the absolute scope of Infinite Unity, through its aspect in absolute multiplicity, and ultimately into the relative scope of finite unity— can mathematics proper even begin.  And it must begin at square ‘one’; literally with the finite unity of the primitive number ‘1,’ which itself is an enfolded implicit infinity, a “bounded infinite” and primitive infinite number as pure-relational counterpart to Spinoza’s “simplest body” which we’ll explore in great depth later.  In the real-world as empirical correlate, this manifestation of our primitive finite unity takes place as the emergence of whatever modification is the bounded object, and set thereof, of the first primitive mathematical operation of counting.  Such manifestation always takes place in a sub-representational arena, beneath and before the representation of mathematics itself, and so in the embryogenesis of the concept of Interface Mathematics as well.  The common tendency to take the immanent pole of this uni-axis—our familiar “zero-dimensional,” “infinitely small” “mathematical point”—as our finite unity and first number must be resisted, because unlike our first “natural” numbers, this point is unbounded, an implicit singularity, and hence an infinite logical-Emptiness, not a finite, form.  It is actually not a unity at all, but an implicit singularity; a single pole of an unbounded polarity, and ultimately merely an immanent positional aspect of Infinite Unity.

266 – 267  Spinoza makes an important clarification to the polarity of the finite and The Infinite, but he places his emphasis between the Infinite and the imagination (i.e. representation), which he sees necessarily as finite, given its modal nature and what we are calling its relative scope in the world of form.ii  This difference, he says in his famous Letter XII on the Infinite, is crucial to untangling the confusions surrounding the infinite (as we will very soon see).  The Infinite is “that, which can be understood but not imagined” and the finite is “that which can [be understood and] also be imagined.”  This is the primary scope distinction between absolute and relative—the fundamental polarity of the Univocity Framework—rendered in terms of the multiple or quantitative aspects of the imagination, and the limitations of the forces of representation.

268  An understanding of infinity is therefore always necessarily a conceptual abstraction based on the mnemonic foundations at the interface of perception/conception, just as every modification is an abstraction from the infinite itself.  But without the imagination and its foundational perceptions to aid in the process of transcending itself, there is no understanding of the finite or the infinite.

269  The finite unity (boundedness or finity) gives rise to duality and only the Infinite Unity (boundless ALL and absolute scope) can be truly ONE and non-dual because it contains and exfoliates all polarity.

278 -279  All representations of The Infinite are necessarily aspect infinities—even the first-phase aspect infinity, the I/T omni-axis—as they draw their infinite nature as representational aspects from The Infinite.  Other aspects include what we could call the second-phase aspects, e.g. the transitive number-line when counting by ones, twos, fours, eights, or recursively self-multiplying into perfect-squares, dividing into the immanent continuity of the rationals, etc.  The abstracted aspects of The Infinite, in these cases would be “square-ness,” “even-ness,” “point-ness,” “line-ness,” “plane-ness,” or “volume-ness” (extension), etc.   Imagine instead of drawing an infinite line (a clear impossibility) that the infinite line is simply abstracted into representational reality, bit by bit, from The Infinite, omni-dimensional continuity which initially contains the possibility of all lines.  The linear aspect of The Infinite appears from the continuity because we put emphasis on it, and it disappears when we let it go.  Once manifest, we can then assign to its intrinsic nondual or trans- rational continuity an arbitrary set of numbers based on whatever set- generating algorithm and relative to whatever transitive scale we choose (e.g. the mathematical ratio generating the set of the Rationals with their Dedekind cuts and intrinsic limits).  …The sense here is that these mental forms of set-generators, omni-/uni- directionality, “linearity,” “planarity” and n-dimensionality are abstracted as singular or grouped directional/positional (relational) aspects from the concept or reality of the Absolute Infinite, but furthermore, the omni-axis enfolds and unfolds the uni- and directional- axes, and hence is a more primitive order in the embryogenesis of the concept of mathematical, ontological and epistemological infinity.

280 -281  In an infinite set, cardinality has a critically different meaning than in a finite set.  It does not denote the number of elements, because an infinite set has no definite, final or actual numberiv of elements.  It denotes, rather, a relation between generative “powers” of set-generation algorithms, as comparable through the one-to-one correspondence seen with the Galilean paradox and vindicated through (the controversial) Hume’s Principle.i  In this simple way we can compare the degrees to which these set-aspects abstract similar or different “powers” of infinity (our quantitative aspect of the absolute scope) from The Infinite, unbounded Emptiness, Cantor’s Absolute Infinite or Spinoza’s God/Substance with infinite attributes.

282   The modal infinite is the order of the fractal, complexity theory and nonlinear and/or recursive functions, and it is also the home of our familiar oxymoron, “finite but unbounded,” and our primitive (finite) number as an enfolded or encapsulated infinite number, in an identity of opposites.  Due to the symmetrical circularity (inverse tautology) of the phrase “finite but unbounded” it easily could be stated in the reverse, “infinite but bounded” because what it really means (given our necessary remapping of finite and infinite to bounded and boundless respectively) is “infinite yet finite” or “bounded yet unbounded.”  The problem here is that the scientists are not aware of the two aspects of directionality pre-fused and conflated (con-fused) in the one Infinity.  They thus quite easily and unknowingly mix them together.

295 – 296  As the levels move down the triune hierarchy, from the absolute through the relative scope aspects, they get progressively more and more relative (as aspects) and finite (bounded), since this triunity represents the cultivated third, or the interface, between the infinite and the finite.  This also corresponds to the move from the trans-trans-bias in the unrecognition of the rational zero (undefined zero in the denominator) to the recognition, in the 17th century rationalists, of immanence in the bounded infinite.   …Though it contains an infinity within itself (and without itself), this bounded infinite with its implicit singularity can act (and must act) as a finite transitive root-unit and interact relative to other units.  This nucleation is indeed how the transitive axes exfoliate (emerge) from the immanence and transcendence of the absolute, though we can clearly see here the

necessary cyclical nature of this non-dual and non-foundational framework rooted in the rootlessness and implicit singularity represented by the uncountable I/T axis.i*  The Cycle of the Infinite is the move from a priori  I/T omni-non-local/-directional boundlessness, to the uni-axis, its implicit singularity and finite unit-interface, and through the holarchical unfolding of number and operation back to function on the I/T axis, making explicit its implicit roots in immanence and transcendence.  We’ll see this in another form in The Binary Cycle of Identities, Axes and Unities (p383).  As we will see, this bounded infinite is the abstract vision-logic description of the Leibnizian “monad” and the Spinozan “simplest body,”i the joint metaphysical and meta-mathematical union of the absolute and the relative; the infinite and the finite.  It is also the previously unacknowledged beginning place of mathematics itself, as it codifies the origin of the number 1 as an implicit infinite number, enfolded into the primitive natural numbers, whose implicit infinity will not be broken open until the Rational numbers and fully with the immanent infinities of the “irrational”  numbers.

203  Despite the wild success of mathematics in its pragmatic roles in science and technology, mankind has long been plagued with paradox and confusion when it comes to infinity and number and the relation of mathematics to physical and mental reality.   …confusions (and indeed con-fusionsii*) hide behind seemingly innocuous common phrases such as “infinitely small,” or “infinitely many,” as well as more formal and rigorous philosophical/mathematical problems such as the paradoxes of the infinite; the philosophical and commonsense problems with the limit and the infinitesimal of the calculus; and the controversy and confusion surrounding Cantor’s notion of the transfinite in modern set theory.  The bulk of these problems, it will be shown, arise ultimately from misunderstandings and con-fusions about the fundamentally polar, nondual and univocal relationship between infinity and number (the finite)—which, as we have seen and will see in much greater detail below, are the quantitative aspects of the absolute and relative scopes, respectively.

204  This current mode of under-developed mathematical understanding (meta-mathematics) is characterized by an extreme level of abstraction, i.e. a paucity of percept in its abstract concepts.  This is due to the academically- common disabling separation of concept from percept—the causal, visual, and hence visceral, connection to the imagination and intuition, and our evolved cognitive interface with reality.We will find that the utilization and expansion of our Vision-Logic Coordinate System (VCS) and Univocity Framework, … readily resolves these paradoxes and confusions.  It grounds the mind in an experiential model for deeper understanding.  This deeper understanding then provides for a powerful resonance between mathematics and nondual-rational, or trans-rational, philosophy…

207  Reality can at once be both mathematical and other than (or beyond) mathematics as the art and science of relation.  It is real relation itself, in this interface of mathematics with reality, that endows mathematics with its power of modeling relation.  This is, in effect, a power of self- similarity between reality and its real echoes into its representational interfaces, and the same self-similarity is at work in the interface between knowledge itself (episteme) and reality (the ontic) interpenetrating and harmonizing throughout Nondual Rationalism, as a whole.

209  Everything we experience in the relative world of form is bounded, in some sense, and made of deeper collections.  These “implicit sets” are, in essence, boundaries around deeper collections, themselves boundaries around still-deeper collections—potentially, logically, and very possibly, ad infinitum.  It is only when the self-identity of abstract categories comes into play that these natural faculties have no ground or traction in difference and begin to slip, leading to paradox.

212  A set is the relative, bounded aspect of the univocal polarity.  The universal set breaks from the container aspect of the holonic set, which in turn is needed to sustain the notion of the empty-set in its tacit breaking from the explicit definition of the set as a collection.  They each break in the opposite direction from relative-scope, holonic percept-concept of the bounded-collection—the whole made of parts, of the implicit holonic set theory—to the absolute scope of the formless and unbounded.

213  The paradoxes of logic and mathematics arise only when we “violate” the Univocity Framework—breaking from the relative scope, unwittingly mono-polizing the unbounded absolute with the relative concepts of sets, and deviating through abstraction from the natural holonic set-logic with its inherently delineated, perceived and conceived holarchical levels (immanent/transcendent relations) of bounded-collections.  …Indeed, as the study of mereology has found, the move from naïve classical set theory to modern axiomatic and “Zermelo-Fraenkel” set theory is essentially a move to reinstall—in abstract, explicit mechanisms—the logic needed to keep track of the levels of sets and to restrict these violations of the sensorial holonic logic already implicit, yet buried under abstraction, in the stratified human mind.

214  The goal throughout the rest of this unfolding of Interface Mathematics is not to replace modern axiomatic set theory, in any of its variations, but merely to inform it with an implicit, intuitive percept-logic of sets; to reconnect the concepts back with the percepts from which they were abstracted into far more general and hence useful symbolic mechanisms.  The end result will be to know the place from which we started more thoroughly, having re-examined and reconnected our roots in perception.

216  We will come to see “infinite sets” more clearly as “set aspects,” or “aspect infinities,” rather than bounded-collections with definitive numbers of elements, magnitudes or “cardinalities.” This new way of looking at things will illuminate and resolve the truth behind this paradox in accord with Galileo’s own reasoning that aspects of the finite, such as relative magnitudes, do not really and fully apply to The Infinite.

217  An infinite set, therefore, reaches into the absolute scope, drawing out the quantitative aspect of the absolute in the unbounded and infinite.  It therefore possesses the quality of the absolute scope in its ultimate inability to relate, one with another.  In other words, an infinite set is not a relative set, but a “finitized” and relativized aspect of the absolute scope.  This is why two infinite sets, such as the squares and wholes—as Galileo intuited and Cantor effectively confirmed, as we will see—cannot properly be compared as to their “number of elements,” and indeed, they actually have no number of elements, but simply the numberless quantitative aspect of the infinite.  Infinite sets can relate one with another, and to finite sets only univocally, as multiple aspects or voices of the one absolute; The Infinite.

226  In the shock and integrating interface between the pre-modern opposite and the post-modern identity, then, we will see emerging again—but as a cultivated third and triune interface—the ancient nondual identity of opposites of the finite and infinite.  It is this identity of opposites that finds its germination in the lost truths of modernity, namely in esoteric rationalism as we find it in Spinoza’s Triune Infinite, also reflected and refracted through Leibniz’s reactionary monadology.  And in this integration will likewise continue to unfold this “wedding of concept and singularity.”  We have already seen it with the immanent/transcendent omni- and uni-axes, and we’ll soon see it more explicitly in the concept of Spinoza’s Triune Infinite and, perhaps at its climax, in the Interface resolution to Zeno’s paradoxes.

235  So now we can systematically say that the countably infinite set of Natural numbers is numerically identical to its subsets, i.e. “self-nesting,” and we can give them all a single number or cardinality, and use that number to perform a kind of arithmetic between them.  And through this transfinite arithmetic, recursively unfolding above us we can “see” an infinite “paradise” of ever larger and larger infinities.  Indeed, not only can we see this (mathematically speaking), but we can wander through it and perhaps use it for practical purposes.

241  Nonduality can be understood in many ways, as we have seen, but the Univocity Framework—the fundamental polarity of scope and its quantitative aspects in the infinite and finite—provide a unique logical and rational clarity on the subject.  From The Lotus Sutra we find, “The concept, often described in English as ‘nondualism,’ is extremely hard for the mind to grasp or visualize, since the mind engages constantly in the making of distinctions and nondualism represents the rejection or transcendence of all distinctions.”(Watson) And from Bede Griffiths (1997) we find, “Advaita (nonduality) does not mean ‘one’ in the sense of eliminating all differences. The differences are present in the one in a mysterious way. They are not separated anymore, and yet they are there.”

242 – 243 Vision-logic equations (VLE) use the symbols and operators of mathematics to simplify and succinctly display the relations involved at the vision-logic level of Interface Mathematics.  They have slightly different rules, which will be explained as they come up, and are used in a much looser fashion than mathematical equations.  They are used more as quick, symbolic/visual metaphors, or maps, illustrating relations, rather than quantitative derivations.  This is largely because of their extensive use of absolute and inquantate infinity,i and its inability to really partake in quantitative operations—operating, as the VLE often do, between the relative and absolute scopes.

246  As we have seen with the Hilbert and Galilean paradoxes, Infinite Unity, or Infinity (as opposed to the common notion of infinity used, say, in the term “infinite set,” which is merely an aspect of The Infinite), when operated upon—divided, multiplied, added to, subtracted from, counted by odds, evens or perfect squares into its multiple quantitative set-aspects—always returns infinity.  This is the nature of the absolute scope that, even when conceived in its multiple aspect of the ALL, it cannot properly be operated upon without leading to paradox or tautologyi*, such as “self-nesting” and the ONE-is-ALL, respectively.  And this is the core principle reconciling the paradoxes of the infinite—an abstract generality which we will flesh out in much more detail.  We can see this principle as another resonating chord in our Ariadne’s Cable, from our first chord, the Principle of Nondual Rationalism; infinite division equals indivisibility.

252  In the Univocity Framework, both the immanent and transcendent poles (along with any other infinity) are aspects of the boundless absolute, The Infinite, or Nagarjunan Emptiness.  Number, on the other hand, conceived as various functions of boundary, is necessarily dealing within the world of form, the relative scope, not pointing outside itself to the absolute.  To be clear, there is indeed Emptiness or infinity within all numbers, forms and boundaries, and this Emptiness and infinity is the source of number and form itself, but this Emptiness or infinity is never ultimately reached or encapsulated in the symbol-system itself.  It is always critically sub-representational, as Deleuze would say.  It is this separation of the absolute and relative scopes that ensures the functionality of the Univocity Framework, and the various forms of the univocal concept of dependent arising.

254  In Interface Mathematics, once that boundary is drawn (or perceived) a number can be introduced into consciousness, or the imagination, to represent, keep track of, and make use of, the bounded object.  Numbers (again as magnitudes) cannot come into being, either causally or experientially, and have no function without at least tacitly representing, or being capable of representing, some real, physically-conceived limit—be it known or unknown.