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.