Tag Archives: infinite

by

GD. Goethean Experimental Observations for 12/06/2012

(noticing the phenomena of sounds arising from within);

Prelude: I have relocated to a big old house near the beach, with a forest across the road. To honour the occasion, we had a fire ceremony at the beach on the evening of the 12th June. After enjoying the beautiful starlit evening we returned to the house, after I have settled my son I settle into sitting meditation on the mat in front of the altar.

I sound the singing bowl, chant Aum and then sit with my double dorje in my hands, contemplating the energetic nexus that it represents, and creatively imagining the dynamic structure which it represents the underpinnings of.

 

I begin to feel the arising of kundalini shakti, gently undulating through my subtle energetic system and up my spine, out through the top of my head with a tingling sensation and into the familiar fountain-like pattern of return to the overflowing pool at the base chakra where the flow continues up the central channel and back out my head. After sitting like this for some time I feel the need to surrender into savasana (corpse pose) however I am also feeling chilled as it is mid winter so I take my body to bed and continue my meditation there.

At this point my body begins to tremble and shake, to vibrate from a deep-down-phenomenon of silent sound arising in the very core of every atom and cell. I note my heart rate increasing, and breathing deepening and lengthening. I am surrendered, all action taking place now is spontaneously arising of its own accord, I am remaining in lucid awareness and observing.

With the deepening of the breath I note an expansion of the abdomen, accompanied by deep creaking and groaning sounds, my chest and throat are also expanding to the sounds of cartilage creaking. My vertebrae are aligning and stretching apart. From a place deep within my skull a secretion of cool, sweet liquid begins to flow into my throat and upper nasal passages. This flow of (?) seems to precipitate an intensification of the phenomenon.

 

I begin to feel pressure stretching me from the inside like a balloon being filled, the air flowing into my lungs seems to penetrate every organ, membrane and cell of my physical body beyond their capacity to contain it in their present shape. I feel like I am being tickled from within, I am laughing intermittently and potent “chemical” tears are streaming from my eyes. My hands move to hover over my sacral and solar plexus chakras, the hands are moving in rapid circles counter rotational to each other, the chakras feel like ‘solid’ balls of whirling energy. The trembling in my body intensifies, my heart rate is still accelerating and my breathing seems to have stopped at a full in-breath. Part of my mind is cautioning me about the possibility of having a heart attack or brain anurism because of the unusual conditions. I consciously make the decision to surrender and trust the experience, even if it means I do in fact die as a result (!).

At this point an impossibly bright yet cool light erupts in both my heart and head (witnessed in the third eye) which expands out to include my whole body. Now all my physical bodily components seem to completely liquify, there is a sensation of dissolving and (hard to describe what) I imagine being like a chrysalis undergoing the metamorphic process. The level of inner sound which is accompanying this event is tremendous yet also very difficult to describe, something like a howling storm perhaps (I will try to find an approximation on mp3 somewhere). My inner eye is registering myriad complex fluidic geometric structures dynamically interacting and energised by seemingly liquid light throughout my field of awareness, all emanating from the core of my being. My physical body feels like it is a writhing mass of system interactions from the cellular level through to the level of organs, tissues and bones. This is a full spectrum immersive experience of something extraordinary!

The complex “body of light” which it appears to my inner eye that I have become is expanding to fill the room, expanding out across the forest, across the ocean and encompassing the whole planet, the expansion continues, waves of energetic intensification continue, I now contain the whole galaxy, and continue to expand and absorb greater and greater spheres of (?) perhaps awareness of consciousness – I am the universal essence in all its forms for a ‘moment’ – it is almost totally overwhelming. I am not breathing and my heart seems to be stuck on full throttle.

Sound begins to emit from my expansive throat, a deep moaning nameless sound. The observational part of my mind is still restless and registering cautions about the possibilities of damage to the physical vehicle. I decide that I need to ‘ground’ the experience and that I can Aum with it as a means of fully embodying that which is flowing through me (rather than becoming fully disembodied by its raw power). So I begin to Aum, low and deep into the lower chakra triad, and then a higher tone, into the heart, and higher again into the upper chakra triad. I slowly begin to reintegrate and re-inhabit my physical vehicle. The toning is re-establishing a breathing rhythm, my heart rate begins to decelerate somewhat.

I find myself glowing and buzzing, it is as if I have just been born, My body feels like a fresh new organism, weightless for a while, tingling and throbbing with light and sound – I seem to be in psychic contact with every mind I have ever contacted, all of you are right here with me, I hear your voices and feel you in me. We are all connected to this vast and fathomless energy source which is still flowing through me, as me (and everything and everyone). My subtle sensitivities appear to have been enhanced, yet so has my capacity to accept being in such a state.

At the time of writing, 5 days later, I am still in this ‘renewed’ state. Speaking with the Goethe track group (16/06/2012) helped me ground the experience a bit more, thank God for these eminently qualified comrades. Many of the images which have sprung unbidden to my mind over the last 20 years are now relating to each other in my mind in ways that they have not done before – so many symbols of a pre-cognised experience which only now is unfolding – the whole concept of time and space is so plastic. And underpinning the whole experiential context – the sound which arises from within – what a revelation.

Conclusion: No amount of theory can adequately describe a knowing which only direct experience can deliver.

The lucid contemplative reflections continue….

by

SZ. To Infinity and Beyond: Tuning and Triuning the Paradox. pp.433-482.

p.434-435. It is the category of organism as indivisible then, which when implicitly misapplied causes this dichotomy of not quite right conclusions we perceive as a paradox. The intrinsic precision of the actuality cannot be reached by the imprecision of the extrinsic boundaries of the category, and so we oscillate between the two closest approximations:

Paradoxes, then, are errors of intrinsic incompatibility in a compartmentalised, categorical and dualistic logically descriptive system. … From a different vantage point, then – e.g. from a more detailed system of categories -we will see that the paradoxes can, and often do indeed disappear. … by explicitly modelling the pivotal detail inaccessible by the categorical framework rendering and generating the paradox.

The category, then, serves a transcendent and transitive (trans-biased) representational function of power over the territory beneath the category – this territory being namely emergent “meaning” and sub-representation. Language or syntax in this sense is thus transcendent to and emergent from semantics or embodied meaning. Yet syntax, in turn, symbiotically modulates semantics from above, serving as a medium or lens through which embodied meaning can be manipulated and further elevated.

So, the way we communicate concepts effects the way the concepts are represented, and those representations in language then also condition the manner in which the communication takes place, as well as the conceptual structure itself?

 

p.439. This is the simplest “answer” to the paradox, which was honed in the nondual traditions themselves. Opposites, such as the opposing conclusions or categories of the paradox, are simply acknowledged as intrinsic elements of each other. Like yin in yang, and vice versa, they are “identical opposites”. Nondual traditions embrace the paradox because they know that the problem is not one merely of representation (illusion or Maya), in a dualistic and oppositional set of categories. The problem sometimes happens then, that in the embrace of the paradox, they can often tend to hold onto it as essential to reality or nondual philosophy itself. The paradox must be remembered as a dualistic function of illusion and perspective, and it must be conceived free from dualistic attachment in order to retain the possibility of seeing it as one, from a different categorical perspective.

p.441. Can reason itself, through the acategorical imperative, reach into the categorical fissures of common-sense – the paradoxes – and find their common-ground in some kind of deeper level of fluidity and emergence – some nondual form of rationality that resolves paradoxes naturally?

p.447. The Infinite has no internal conceptual relation or multiplicity which can be abstracted free from the derivative relation of the aspect infinite or its modal interfaces, which are its conceptual modalities – its senses. Abstraction into relation is this interface.

p.460 … polarity is the most general function of the world of conceptual relation, which of necessity includes the world of logic. In his [Zeno's] paradoxes of motion, we will see plurality expressed through the infinite divisibility of change, and through the same underlying form the simultaneously infinite division of time is demonstrated to be the equivalent of the indivisibility of eternity manifesting in the Zenonian instant. Motion, then, is not so much denied, as demonstrated to be indivisible, continuous and ultimately ONE eternal and instantaneous Unity – the Parmenedian Being-now. And this is naturally in conjunction with our PNDR “infinite division equals indivisibility”, and its Ariadnian resonance into the chord of “infinite determinism equals indeterminism” and its eternal-NOW.

p.463 If we “pulverise” the category of “thing” upon the exoteric ambiguity hidden within it, we can divide it into two categories; individuals and dividuals (e.g. a-tomoi and holons) Zeno’s argument merely demonstrates, and purposefully so, the impossibility of the absolute individual (a-tom), with respect to the concept of size, and thus indirectly for the implicit holonic logic of sets, with its implicit unity in the ONE-ALL.

p.469. Verelst demonstrates that all of Zeno’s paradoxes of motion (PM) – including even the stadium paradox – can be subsumed under the timeless and instantaneous form of the general paradox of plurality (PP), rendered into mathematical rigor, above, and explained in her paper in the formal language of domain theory.

It is the uncountable infinite of the ignored PP which already exists as the actual infinite underlying all of the PM. And further, it is divisible movement that takes on the time-ordered, stepwise function of the potentially infinite, never able to reach the immanent pole of the actually existing omni-axis within which motion acts.

p.476. As we have seen throughout mathematics and the conceptual embriogenesis itself, immanence breaks the trans-bias to effect a transcendence into new operations and identities. Likewise Zeno’s naked paradox radicalizes immanence to break the transcendent bias of dualistic representation – i.e. diction – shocking us, generation after generation into various and sundry attempts to refute him. All the while, in this very interface – in this refutational dance with Zeno himself – we move closer and closer to operationalizing and vindicating the polarity and truth hiding in the naked core of his ubiquitous paradox.

It strikes me here, that with enough of a stretch of the mind, the resolution of these paradoxes will occur spontaneously. It makes me think of the sages saying that when we realize the real, the actual, in its simplicity, we will tilt our heads back and laugh at the sky! I believe we will begin giggling soon…..

by

SZ. The Holarchical Unfolding of Number and Operation. pp.308-401.

Phase TWO: unit-closure and the finite unit-identity.

p.310. … the very first number, the finite unit-identity, is born from a coupling of an identity and an operation in the differentiation-and-integration of the embriogenesis of the concept.

p.311. There is, then, a critical Heralcitian polarity of identity and operation; noun and verb; stasis and flux, implicit from the very beginning of this embriogenesis of the concept of mathematics and number, and we will find it recurring throughout … and as we have already seen in the Univocity Framework.

The primary number ’1′, as we saw in the Cycle of Unity, appears as a boundary between immanence and transcendence, and as the basic modal infinite. … given that each unit-identity is composed itself of a bounded infinite, the n each composite number is a primitive infinite number, whose infinity is simply encapsulated in the unit-identity itself.

Phase FOUR: transitive interfacing with the I/T axis.

p.326. The Immanent/Transcendent axis, we have seen, like all aspect infinities, is prior to mathematics and operability. Like the imaginary absolutised categories of the Platonic Forms (or even our natural “numeric identities” and digits), it is therefore, by definition, ultimately abstract, invisible, unmodelable and inoperable, in and of itself. It must be accessed, visualised, mobilised and actualised bit by bit through the real-illusions, forms, interfaces and operations of the relative scope, beginning with the operations of transitivity, in the same sense that Substance must manifest itself – to itself, and through itself – as pure emergent and unpredictable potential, in actual modification, nucleation, evolutionary agglomeration and complexification (Emptiness in form).

[from footnote p.326] It is my current assumption and discovery that the unconscious accidental conflation between these two unknown operational axes causes perhaps all of the known paradoxes in ancient and modern Western meta-/mathematics, as we have already begun to see. We will address many more of them as we continue to explore this holarchical unfolding of number and operation.

p.329. … a triangular multiplication table is inherently more efficient than a square one, which also demonstrates, along with other concepts that we’ll explore, that powering is not inherently “square” in nature, but can be more economically and optimally seen in “triangling” and “tetrahedroning” as opposed to “squaring” and “cubing”.

p.331. … violation of closure at the transition from the integers to the Rationals is of such importance in mathematics that it is often called simply “violation of the closure property” itself, and this is because it violates the unit-closure, and pulverises the fundamental category of mathematics, the unit-identity (and the agglomerative numerical-identities or the integers) that began the holarchical embriogenesis of number and operation in the first place.

p.333. … when we divide past unit-identity we break the trans-biased unit-closure of holarchical recursive agglomeration and enter into infinite immanence as though we might search to redefine the unit scale. This immanent infinity gives rise to the infinities between all finite numbers on the continuous or rational number-line. It is the land of the undefined logical-Emptiness that must be mapped in terms of the defined, and thus we find the Rational numbers and the “Decimal expansions”, operating upon the new I/T axis yet expanding from either direction in relation to, transcending-and-including, the transitive axis of integers.

p.339. With infinite volumetric extension again inherent and enfolded in the definition of magnitude and primitive number itself – and thus accompanying all instantiations or ideas of quantitative value, especially concerning PHYSICAL parameters – there can be no more “infinitely small” point-particles the draw us into the labyrinth of the continuum, or into Zeno’s infinite regress of representational measurement, only to be empirically “renormalised” later into the finite, volumetric- or extension-derived physical constants of particle physics. … The extensionless point – as the implicit singularity of an immanent/transcendent uni-axis or as the positional aspect of The Infinite – can only denote position (a ‘terminus’, as Leibniz called it), after the fact, not physical extent or magnitude. Continuity and Infinite Unity is not composed of immanent infinities, but they are abstracted out of it as positional aspects.

Immanent zero, conceived as a limit, is instantaneously squeezed out of transitive extension by the categorical realisation that “zero-size” is absolutised into an oxymoron and, like “absolute nothingness, it is a self negating concept. Thus the immanent “zero” behind every infinite division becomes the abstract inward pole of the infinite immanent axis beyond transitive extension and numeration. It is the inverse of “infinite in extent” so we call it the “inverse infinite” or “immanent zero”. And as a pole of an infinite polarity or VL(vision logic) -axis, it is absolutely unreachable and boundless.

p.354. If we want a four-dimensional geometrical and modelable system, then we must redefine dimension and orthagonality away from 90-degreeness.

p.356-357. Synergetic geometry is inherently four-dimensional and so the “imaginary axis”, with it’s ad-hoc implicit axiom of perpendicularity, is no longer needed. From this vantage point, the term “imaginary” is vestigial, so it drops off.

.. powering or exponentiation – and the system of number and operation itself – is not intrinsically rectilinear or cubic, and also that, according to Fuller’s view, rectilinear analogs for 2nd and 3rd powering are sub-optimal, relative to his synergetic geometry. … it is inaccurate to say “squaring” for second powering. Squaring should indeed be reserved for operations in geometry that intrinsically deal with squares. Powering is not such a case.

p.359. … there are very strong structural reasons to consider that the triangle, not the square, is inherently planar, mainly because three vertices is the minimum needed for any plane, and also that the triangle is the only polygon with inherent structural stability.

p.368 [footnote] Open Question: What is the significance of the fact that the first two irrational numbers are the square roots of 2 and 10? Does it show the limits of a more rational, or immanent/transcendent base-nine system underlying the transitive base ten, as Bucky intimated.

What has this to do with the symmetrical Scheherazade numbers?

p.370. … like Cantor’s “transfinite numbers,” irrational numbers are aspects of the infinite interfacing with the finite function that generated the number; they are bounded aspect infinities, not numbers as finite boundaries.

At this point I am re-reading the algebaic representations of the concept space, simply because my mind is sliding off this form of  explication and I would really like to grok the math on its own terms. The diagrams are helping with integrating the mathematical terms with the experiential means by which I make sense of these ideas – but the cartesian graphology and algebraic referents are doing strange things to my cognitive structures…. ~:)

 

by

SZ. Pt.2 Interface Mathematics. pp.259-307.

 

… in the univocity framework, the ONE-ALL of the omni-axis is an absolute scope polarity, and therefore a univocal tautology, where infinite omni-local immanence is infinite transcendence and the ALL is already contained in the ONE , and vice versa. In this absolute ineffable tautology we can oscillate endlessly, moving everywhere at once and thus nowhere … except for the nondual realisation of this identity itself.

p.261. There is, at this point in the embriogenesis of the concept of mathematics, only the pre-differentiated Infinite, absolute boundless, formless, logical Emptiness or absolute Unity, abstracted into an undifferentiated “multiple aspect”, the infinite multiplicity and continuity of the ONE-ALL.

From this pre-differentiated multiple aspect in the ALL we begin the embriogenesis of operation and number. But only through a “breaking of symmetry” – from the absolute scope of Infinite Unity, through its aspects in absolute multiplicity, and ultimately into the relative scope of fimite unity – can mathematics proper even begin.

pp.266-267. (In Spinoza’s view) 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.

In Spinoza’s view, all images or representations are part of the relative and finite (or indefinite) imagination. This would then include perception and conception, while the reality of this necessary representational illusion of sensation and other forms of the imagination is our qualitative connection, our interface with, and as, reality (and specifically a finite unity thereof).

The best we get with the imagination is the indefinite, not the infinite.

Therefore, as the rationalists correctly noted, the abstract understanding, coupled with the intuition, can grasp things (at least relatively so) of a general and symbolic nature extending beyond, or transcending and including the capabilities of the detailed imagination, including both objective and subjective forms such as perception.

p.268. But without the imagination and its foundational perceptions to aid in the process of transcending itself, there is no understanding of the finite or of the infinite.

A note on Fuller’s Finite Absolutism. … as soon as a boundary is introduced for the extent of Universe, its magnitude is surpassed by the necessity of an outside in relation to an inside.

p.270. The Triune Infinite: Interfacing Emptiness.

According to the evolution and embriogenesis of the concept, polarities generally differentiate and integrate to triunities, or cultivated thirds. What then is the cultivated third of the polarity we just explored between The Infinite and the finite, or more properly between the quantitative aspects of the absolute and relative scopes?

… it is fascinating to note that many thinkers, for whatever reason, have independently come to the same general conclusion of the three types or degrees of infinity. … there may be more than coincidence at work here in this relative interface of the number three with the absolute scope or the divine.

p.271. … essentially, Cantor’s three infinites can be expressed as absolutes, Ontic and Mathematical (this last category of mathematical infinities is merely a subset of the epistemic or representational forms of the infinite (aspects), i.e. mathematics as the representation (art/science) of pure relation.

p.274. Order 1; The Absolute Infinite.

The 1st order Infinite is “The Absolute,” the ALL is ONE, the “Boundless ALL”, our absolute scope, Emptiness and Infinite Unity which is not ‘one’.

p.275. The imagination is always exhausted in a finite amount of time and therefore can only ever reach a limited, indefinite depth of extension (transitive), division (immanence) or time.

Order 2: The Aspect Infinite.

The Aspect Infinite is the second order infinity and it derives its infinite nature directly from the first as unfolding aspects, perspectives, infinite modes of being or mental abstractions from it. The Aspect Infinite mediates as a cultivated third, or a triune-interface between the first and last orders of infinity, (footnote: it is important to note that the absolute and relative scopes are not locations, or scales of interaction in a relation or reduction, one to the other. They are, rather, the basic conceptual contexts from which the mind attempts to make sense of the world and itself. Indeed, the absolute scope only comes in to the conceptual field of view when the relative scope attempts to relativise itself in order to find its context in its identical opposite. And thus the bounded relative scope must find its basis and meaning in the boundless absolute scope and the concept comes full circle to grasp its genesis in the pre-conceptual, and the finite in the infinite.

p.279. The linear aspect of the Infinite appears from the continuity because we put emphasis upon it, and it disappears when we let it go. Once manifest, we can then assign to its intrinsic nondual or transrational continuity an arbitrary set of numbers based on whatever set-generating algorithm and relative to whatever transitive scale we choose …

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 embriogenesis of the concept of mathematical, ontological and epistemological infinity.

p.281. Order 3: The Modal (or Bounded) Infinite:

The Bounded Infinite is the third and lowest order of infinity as it deals with the unbounded within the bounded.

The modal infinity is a 2nd order derivative from the Absolute Infinity, and it unfolds from the confluence of the two main aspect infinities: the I/T (omnidirectional) and the transitive (uni-directional) derived from our finite unity as a frame of reference. The modal infinity is the interface between the immanent and transitive aspects or directionalities, and can thus be seen as a subset of the 2nd order…

p.282. We have found only unbounded dimensions, because beyond every boundary we have always found more dimension, where we are capable of finding anything at all. Indeed, a boundary cannot even be found until it is surpassed, exposing the fundamental polarity of the infinite and finite.

p.283. Everywhere man looks he finds the boundlessness of the indefinite, the interface between The Infinite (both within and without him) and the finite.

p.287. … we can also take Spinoza’s definition of The Infinite and finite generally in termsof the polarity of Substance and its Modifications, respectively, and it’s Cycle of Unity, within which substance turns around its modes and vice versa. Tha which is modified is, in some sense, necessarily bounded and has magnitude (e.g. size, length, width, duration) reletive to other modifications; modification is necessarily in the relative realm of form in identical opposition to the absolute realm of Emptiness.

p.291. Since we are dealing with the absolute scope, and therefore necessarily with the philosophical and pre-mathematical, rather than the mathematical notions of infinity, and given that infinity in either case is not a number these (vision-logic equations) VLEs are not rigorously mathematical, but rather meta-mathematical, logical or relational, with their own peculiar logic.

p.296. … this unbounded infinite is th abstract vision logic description of the Lebnizian “monad” and the Spinozan “simplest body”, 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.

p.297. Boundary and Dimension:

… beyond the infinite representationality of dimensioning systems, there is real, finite and infinite, sub-representational, a priori volumetric extension and it is only quantified (differentiated and integrated in the embriogenesis of the concept) a posteriori with the use of dimensions ad systems thereof.

 

by

SZ. Pt.2 Interface Mathematics. pp203-259

p.205 Modern meta-mathematics and hence mathematical physics are “non-modelable” and in effect, pre-rational. They do not function in harmony with the perceptual foundations of the human mind, and hence cannot be modelled effectively in the imagination – and as a subset of self-similar reality, the mind should naturally be able to echo itself into a reflexive understanding of its own unified or nondual substrate as dependently arising in both external (objective) and internal (subjective) reality.

p.206 Mathematics is merely a tool for abstracting and interfacing the quanitiative/multiplicative aspects of the finite and infinite at the most abstract and general level of relation. This depth of generality in the embriogenesis of the concept gives mathematics its power and extreme applicability fro modelling relations in virtually all fields of conceptual, exploratory or empirical endeavour.

p.207 It is real relation itself, in this interface of mathematics with reality, that endows mathematics with its power of modelling 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 harmonising throughout Nondual Rationalism as a whole.

pp.209-211 This natural, intuitive or implicit logic – founded upon the human perception of sets as bounded collections – is encoded, if abstractly and incompletely, in the “part-whole axiom” of classical mathematics which states simply that “every set is greater than its proper subset”. … 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.

Is a set a collection or a container? If it is a collection, or a “collection of objects”, as is commonly claimed, then the “empty set” (as I will show in more depth below) is inconceivable, unimaginable and illogical, and merely implies the absence of a set; a non-set. If a set is a container, on the other hand, which is needed to make sense of the non-collection of the empty-set, denoted, for example, by the notation {}, then an – such as that used extensively by Cantor and modern axiomatic set theory – is infinitely illogical and inconceivable because a container cannot sensibly be “infinite in size”; a boundary, by definition, cannot be boundless. An infinite container, then, would likewise be a non-container.

All sets as categories are now conceived in freedom from natural holarchy, causing a naïve disruption of the connections of percept taken for granted and forgotten in the movement into concept. These lost connections in naïve set theory are the bonds symbolically or axiomatically replaced in the newer axiomatic versions of set theory, and, as we will see, in the study of mereology, literally the logic or study of parts.

p.212 Neither the universal set nor the null set, are “proper” sets, in the implicit holonic logic of sets as bounded-collections, or part-wholes; they break away from the relative world of form into the absolute realm of the formless. They possess the absolute and unbounded aspects of affirmation and negation, respectively.

p.219 The Nondual-Rational and Empirical Embriogenesis of Mathematics.

p.223 The Polarity of the Finite and Infinite.

The finite and infinite are the polar aspects of boundedness and unboundedness fro the relative and absolute scopes, respectively. And in terms of pure-relation, they are the quantitative aspects of our polarity of scope – the polarity and “contra-diction” at the very foundation of quantitative reasoning itself, because our primitive notions of number correspond precisely to boundary, … The absolute scope, then, deals in the generally ineffable aspects of unity and infinity, and the relative scope deals in the polarities and multiplicities of the finite.

p.228-229 Mathematician Dr. Reviel Netz “… the defining property of infinity today is that a set’s cardinality [its number of elements] is equal to the cardinality of some real subset of that set.

The new definition of the infinite, like that of the finite, is essentially a codification, incorporation and encapsulation of the Galilean part-whole paradox, rather than its resolution. We have rightly accepted the paradoxical nature of the infinite, in its identity to the finite, but in simultaneously discarding its opposition we have yet to understand its nature. We have yet to tune and triune the paradox into a truly nondual identity of opposites, and thus into a triune interface of inter-expression.

The paradox has not been solved by this definition, but has now simply become the definition, subsumed under the identity of the concept or category of the set.

p.230 Indeed the human mind can reason about and, we will see, understand the infinite but this understanding simply hasn’t made its way into philosophy yet.

p.235 Cantor’s project then, like Zeno’s, is not incorrect, but simply incomplete, being subsumed now by the identity of the category.

p.236 In short, only for mathematicians, and essentially only in the language of mathematics, has the paradox been solved. For the bulk of us still wrestling with the angel of understanding, we remain stuck in a head-lock with the same old troubling paradoxes – still confused by “self-nesting” violations of the implicit holonic logic of sets and the relations between the infinite and finite – as between God and man.

p.237 Mankind as a whole seeks not merely abstract, mathematical and syntactic answers, but also to grasp imaginable and visualisablesemantic answers; answers that make sense to the human mind at all levels, from percept to integrated concept.

p.238 Merely accepting the paradox into the hermetically sealed axiomatic layer of tacit assumptions – with no explanation of its core polarity whatsoever – gives the common impression that the problem has been solved.

p.239 Principle 6: The Pearl Principle of Axiomatic Encapsulation.

The tendency to encapsulate irreconcilable paradox into principle; dilemma into dictum; enigma into equation; nonsense into nomenclature; or ambiguity into axiom – in order to reduce the irritation and stress from repeated and constant contact with the unknown.

How do we take the knowledge that this is our tendency and continue to work toward ever deeper clarity without falling into using it? especially the nomenclature clause!

Addressing this, Joel and Glisten talked about it in this recording

Call with spinbitz, Mon Apr 02 2012, 08:08:40

p.242 So, with the preliminary distinction between the absolute and relative scopes, and the infinite and finite aspects of unity, the leap of intuition from bounded and relative notions of unity in the multiplicity of everyday objects, to the Infinite and univocal “ONE is ALL” in the nondual, becomes more fully explicated and much more easily replicated or communicated at the cognitive and logical level.

p.243 Infinite unity includes ALL, period, so there can only be ONE. This simplest of ideas is easily forgotten and the words blurred into new meanings and confusions.

Infinite Unity, being unbounded and absolute, cannot participate in relational, and hence mathematical operations. Without boundaries, it cant relate. It is ineffable. It cannot be added to, subtracted from, divided by or multiplied, because by definition it unfolds and enfolds ALL in existence; there can be nothing to add to everything and nowhere else to subtract it to. There is nothing else to relate Infinite Unity to and no outside perspective, operation/operator or implicit hidden set, from which to withdraw or transfer any arbitrary qualities. The “ONE” is not a number, it is “inquantate”, because Infinity itself, the boundless ALL, is not a definable or “boundable” magnitude.

p.245-246 Anywhere on the immanent-teanscendent axis that we place the solidus boundary of the one, we still have infinity on both sides, and thus – no matter what the scale of the volumetric solidus/viniculum – it is always exactly in the middle, yet the infinity has not been quantitatively decreased by half even if you discard one half for the other.

This division of unity into polarity …is therefore not properly mathematical or quantitative, but pre-mathematical and pre-quantitative or meta-paradigmatic; at the vision-logic level of meta-mathematical cognition (e.g. the VLE). It is merely a percept integrated conceptual exfoliation of a possible intrinsic relation, or a way of conceiving the polarity between The Infinite and finite unities. It is this vision-logic, nondual-rational, or trans-rational polarity within which the mind can cycle. It is also, essentially, the meta-mathematical abstract rendering of the union between God and man; the absolute ineffable Infinite Unity and the relative, effable finite unity as the solidus-viniculum between infinite immanence and transcendence.

p.249 Buckminster Fuller’s most famous and misunderstood maxim, arguably, is “unity is plural and at minimum two”. When properly unpacked into the imagination, this enigmatic phrase reveals an important yet deceptive;ly simple duality which opens the way to grasping the fundamental difference (and polar integration) between infinity and number. This concept is also essential to Operational Mathematics because it is the essence of finite volumetric extension and relation and hence is key to sensorial modelability. “A system, says Bucky, is a ‘conceivable entity’ dividing Universe into two parts: the inside and the outside of the system”.

p.252-253 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,

… the distinction between infinity and number continues our visual polarity between the formlessness of Emptiness and the boundedness of form. The abstraction is given an experiential and sensorial grounding in the causal-logic of extension and, as we will see, this allows it to retain a connection with the causality which, through billions of years of evolution, has informed the innate and powerful imagination of mankind, in its holarchic navigation between agency and communion.

by

SZ initial impressions.

SZ 1. p.71 JM quotes Stewart regarding Leibniz

“Leibniz would censor himself of thoughts that the world was not ready
to receive.”

I love the idea of treading the fuzzy boundary between non-dual
rationalism and empiricism – I wonder how many thoughts WE will generate
that “the world is not ready to receive”?

p.78
Transcending and including (recognising and accounting for) this transcendent-bias, to get to the non-duality and neutrality, the open ground of rationalism, this must be seen as a true polarity, and the qualitative differences between the values that seem so absolute through the active frame of reference of the transcendent numerator, must be seen, in a sense, as relative to the active frame of reference of the inherited interpretation. In other words, the difference in quality between immanent and transcendent positions are a function of the active frame of reference and order of operations of division. The numerator’s frame of reference is active as the determinant of value, and the denominator’s is passive as the context for its meaning.
p.84.
Mathematical ratios, and hence rational numbers, have no intrinsically negative values. Try as we might, but they can’t be reached by either multiplication or division.ii* And hence the “negative infinity” of the Rationals (to entertain the transcendent-bias further) is in the immanent direction, and on the positive side of the integer-axis, ever toward the “unlimit” of zero.iii The rational zero as infinity, then, will remain undefined until the transcendent bias is lifted and we can explicitly and operationally see the infinite axis, and indeed the infinite interminable pole “terminating” at the immanent zero, the inward pole of the positive infinity of the Rationals.
p.86
Fundamental Principle of Nondual Rationalism:
Infinite divisibility equals indivisibility.
Infinite divisibility necessitates that there can be no fundamental or absolute division because there will always be a deeper level of divisibility, and hence, with infinite divisibility the absolute is fundamentally indivisible. This, we will see, is a nondual codification of the truths underlying Zeno ’s general paradox, which itself was not so much refuted by the modern mathematics of the continuum, but vindicated by it. Cantor showed that Zeno’s paradox is a natural feature of the mathematical continuum. Infinite divisions are infinitely small , and any possible gap between two divisions is filled with an infinity more , thus leaving no gaps in the continuum.
p.90
Spinoza’s whole system was founded on immanence in transcendence (yin in yang), with the univocal unity of Substance, whose essence is expressed through its attributes, as the equals-sign between the indivisibility of mind and the infinite divisibility of matter.
pp.93-94
The idea and Cartesian goal of absolute certainty is itself, however, an absolutization of the transcendent or emergent possibilities of the mind, at the expense of the dependency of representational mind itself on the unknowns and unknowables of immanence, and its immediate environment. We simply do not know that the immanence in bodies is incapable (nondually or non-reductionistically) of manifesting the emergence of higher levels of intelligence and representation in human and into our categories of mind and matter. We have no empirical knowledge that a nondual dependent arising, or “symbiogenesis of subject and object” is impossible, and therefore we have no recourse or reason to absolutize the concepts of Mind and Matter into an irreconcilable pair of categories.
p.96
This identity of opposites between the immanent and transcendent is a crucial key to Nondual Rationalism, and it can be seen as a corollary—a correlating and strengthening strand—to the original Ariadne’s thread, the Principle of Nondual Rationalism.
p.99
… the interface of the finite (relative) and infinite (absolute) gives rise to the indefinite.
pp.100-101
The last century of ontology and epistemology, coupled with the discoveries of quantum and complexity science, has rendered moot the lingering, peculiarly medieval, fear of the infinite regress and thrown it into positive relief. Instead of viewing the problem from the negative point of view of looking backward or inward to find an origin in space, scale or time, and fretting when these hypothetical beginnings can’t be found, the problem is turned on its head. If there are no origins to begin with—in other words, if, contrary to our tacit predispositions, the universe is eternal and infinite (as conceived in Buddhist nondual schools of cosmology), in both depth and span, then the search for an origin is itself a false problem engendered by the false premise that such an ultimate beginning or ultimate foundation necessarily exists. The problem of the regress itself is seen to be merely a projection of a regressive, point of view on the infinite or the unbounded nature of reality.

I can sense a gradual revelation of the context in which the categories of contents of consciousness arise, the compass being set here for deep exploration of HOW it is that the universe arises rather than “from where” is enough for this sailor to weather the storm of comparative philosophy that this treatise promises to be.