All the number types - binary, prime, natural, integers, rational, (algebraic) irrational, transcendental, complex and transfinite (in both positive and negative and real and imaginary forms) have well defined meanings from the conventional mathematical perspective (in the accepted analytic sense). Equally they have (unrecognised) holistic meanings with an intimate relevance for the scientific integral understanding of all possible stages of development.
I will just briefly clarify the alternative holistic meaning of the number types (which was the subject of "The Number Paradigms").
We start with the binary numbers 1 and 0.
In holistic terms these numbers are ultimately identical representing the equality of form (1) and emptiness (0). Development commences from the position where they are identical in a totally confused manner (i.e. where neither differentiation nor integration of structures has yet taken place).
So the journey is from this starting confused state to the ultimate mature state of unity (where the differentiation of phenomena has become so refined that they no longer even to appear to arise). In this way this highly dynamic experience of form as relative, becomes inseparable from the absolute present as emptiness.
Perhaps the most exciting breakthrough at this time was the clarity I achieved with respect to the holistic notion of prime numbers. This bears close relationship with the notion of primitive in psychological terms where both conscious and unconscious aspects are still largely confused.
This was to later lead me to the realisation that the study of the ultimate nature of the primes (e.g. with respect to the Riemann Hypothesis) could not be properly undertaken in the absence of both distinctive conscious and unconscious modes of understanding. And as Conventional Mathematics is formally defined in a merely rational (conscious) manner, this poses insuperable difficulties for successfully tackling this issue in the standard manner.
In conventional terms, the natural numbers represent unique combinations of prime factors. In a corresponding holistic sense, natural type experience arises from an underlying psychological base (of primitive instinctive behaviour). However as the infant slowly learns to relate these instincts in an organised manner, the conscious element of experience gradually becomes consolidated.
Thus highly organised experience of objects would require ever greater degrees of conscious specialisation!
The integers involve negative as well as positive natural numbers. In like manner the holistic equivalent entails both positive and negative elements in the growing ability to clearly separate the (external) world from the (internal) self.
So development so far has taken us through the "lower" levels of Band 1, where the conscious becomes ever more clearly separated from the unconscious aspect.
Then at Band 2, full specialisation of rational ability now unfolds. This corresponds in holistic terms with the growing tendency to fragment phenomena into parts through analytic ability, which coincides with the analytic equivalent of rational fractions.
The "higher" levels (Band 3) then leads to the holistic notion of irrational numbers.
In fact the great problem that the early Pythagoreans faced was their inability to satisfactorily provide a holistic explanation of the notion of an irrational number!
Thus a decisive breakdown occurred with respect to the historical development of Mathematics, which increasingly came to abandon holistic notions in favour of a merely reduced analytic meaning.
So in my approach, I showed how the holistic notion of the (algebraic) irrational numbers is closely associated with the nature of Level 1 (Band 3).
I then showed how the notion of a transcendental number (which is also irrational) leads to a greater degree of subtlety. In fact just like π as the most famous transcendental number entails the relationship between a line diameter and its circular circumference, equally the holistic equivalent entails the corresponding dynamic relationship as between both linear (dualistic) and circular (nondual) meaning (as the centre which is common to both).
I also dealt in some detail in this book with the holistic meaning of the directions i.e. positive and negative, real and imaginary and the special limiting complex directions (that are represented as null lines).
In fact this book even went further in showing how the nature of the sub-levels (of each level) could themselves be encoded in a holistic mathematical manner.
In particular, I was anxious here to explain why development in our present culture (especially in mathematical and scientific terms) tends to plateau so strongly with the specialised expression of mere rational understanding i.e. Band 2.
Generally speaking in development at each level, concrete (part) perceptions are first to emerge; then all going well, the next stage will lead to the corresponding emergence of the more general formal (whole) concepts. Finally the interaction of both formal and concrete will take place paving the way for transformation to the next level.
Now the key to understanding the dynamics here is that - properly speaking - part and whole are quantitative as to qualitative (and qualitative as to quantitative) with respect to each other. So in distinguishing concrete and formal sub-levels, I used the Type 1 and Type 2 notions of number respectively to represent both!
This in turn implies that a certain creative fusion of conscious and unconscious aspects is necessary (i.e. of the Type 1 and Type 2 aspects of number) at every level to enable the successful transition to the next level.
Now with the various levels of Band 1, though conscious and unconscious aspects are still confused to a degree with each other, sufficient mutual interaction takes place to enable successful transition between its main levels to take place.
However at Band 2, understanding now becomes so consciously specialised in a reduced manner with only the Type 1 aspect of number recognised, that for most individuals, insufficient interaction as between conscious and unconscious occurs that can enable the radical transition to the more intuitively inspired "higher" levels.
Therefore in our culture, though the unconscious is still necessarily involved, its role is largely reduced to merely serving the needs of conscious type appreciation.
Now there were still many flaws with my approach at that time (especially in terms of adequately explaining the most advanced levels).
However the basic premise - and indeed all the basic insights - were well founded.
In other words the holistic mathematical approach serves as the appropriate basis for an integral scientific approach to development (which potentially entails the entire spectrum of stages).
Also this approach highlights the truly complementary nature of both physical and psychological processes (in horizontal, vertical and diagonal terms).
This entails therefore that just as we have many possible levels of psychological understanding, equally we have many possible levels of understanding with respect to the physical world.
So there is not just one valid scientific approach (that equates with Band 2 of the Spectrum)!
There are in fact many other more refined scientific approaches, that apply to the more advanced Bands. Therefore comprehensive understanding requires a truly multi-layered (i.e. multi-level) scientific approach.
So for simplification, I outline three major approaches to Mathematics and Science; 1) analytic 2) holistic and 3) radial, entailing the interpenetration of analytic and holistic.
I will just briefly clarify the alternative holistic meaning of the number types (which was the subject of "The Number Paradigms").
We start with the binary numbers 1 and 0.
In holistic terms these numbers are ultimately identical representing the equality of form (1) and emptiness (0). Development commences from the position where they are identical in a totally confused manner (i.e. where neither differentiation nor integration of structures has yet taken place).
So the journey is from this starting confused state to the ultimate mature state of unity (where the differentiation of phenomena has become so refined that they no longer even to appear to arise). In this way this highly dynamic experience of form as relative, becomes inseparable from the absolute present as emptiness.
Perhaps the most exciting breakthrough at this time was the clarity I achieved with respect to the holistic notion of prime numbers. This bears close relationship with the notion of primitive in psychological terms where both conscious and unconscious aspects are still largely confused.
This was to later lead me to the realisation that the study of the ultimate nature of the primes (e.g. with respect to the Riemann Hypothesis) could not be properly undertaken in the absence of both distinctive conscious and unconscious modes of understanding. And as Conventional Mathematics is formally defined in a merely rational (conscious) manner, this poses insuperable difficulties for successfully tackling this issue in the standard manner.
In conventional terms, the natural numbers represent unique combinations of prime factors. In a corresponding holistic sense, natural type experience arises from an underlying psychological base (of primitive instinctive behaviour). However as the infant slowly learns to relate these instincts in an organised manner, the conscious element of experience gradually becomes consolidated.
Thus highly organised experience of objects would require ever greater degrees of conscious specialisation!
The integers involve negative as well as positive natural numbers. In like manner the holistic equivalent entails both positive and negative elements in the growing ability to clearly separate the (external) world from the (internal) self.
So development so far has taken us through the "lower" levels of Band 1, where the conscious becomes ever more clearly separated from the unconscious aspect.
Then at Band 2, full specialisation of rational ability now unfolds. This corresponds in holistic terms with the growing tendency to fragment phenomena into parts through analytic ability, which coincides with the analytic equivalent of rational fractions.
The "higher" levels (Band 3) then leads to the holistic notion of irrational numbers.
In fact the great problem that the early Pythagoreans faced was their inability to satisfactorily provide a holistic explanation of the notion of an irrational number!
Thus a decisive breakdown occurred with respect to the historical development of Mathematics, which increasingly came to abandon holistic notions in favour of a merely reduced analytic meaning.
So in my approach, I showed how the holistic notion of the (algebraic) irrational numbers is closely associated with the nature of Level 1 (Band 3).
I then showed how the notion of a transcendental number (which is also irrational) leads to a greater degree of subtlety. In fact just like π as the most famous transcendental number entails the relationship between a line diameter and its circular circumference, equally the holistic equivalent entails the corresponding dynamic relationship as between both linear (dualistic) and circular (nondual) meaning (as the centre which is common to both).
I also dealt in some detail in this book with the holistic meaning of the directions i.e. positive and negative, real and imaginary and the special limiting complex directions (that are represented as null lines).
In fact this book even went further in showing how the nature of the sub-levels (of each level) could themselves be encoded in a holistic mathematical manner.
In particular, I was anxious here to explain why development in our present culture (especially in mathematical and scientific terms) tends to plateau so strongly with the specialised expression of mere rational understanding i.e. Band 2.
Generally speaking in development at each level, concrete (part) perceptions are first to emerge; then all going well, the next stage will lead to the corresponding emergence of the more general formal (whole) concepts. Finally the interaction of both formal and concrete will take place paving the way for transformation to the next level.
Now the key to understanding the dynamics here is that - properly speaking - part and whole are quantitative as to qualitative (and qualitative as to quantitative) with respect to each other. So in distinguishing concrete and formal sub-levels, I used the Type 1 and Type 2 notions of number respectively to represent both!
This in turn implies that a certain creative fusion of conscious and unconscious aspects is necessary (i.e. of the Type 1 and Type 2 aspects of number) at every level to enable the successful transition to the next level.
Now with the various levels of Band 1, though conscious and unconscious aspects are still confused to a degree with each other, sufficient mutual interaction takes place to enable successful transition between its main levels to take place.
However at Band 2, understanding now becomes so consciously specialised in a reduced manner with only the Type 1 aspect of number recognised, that for most individuals, insufficient interaction as between conscious and unconscious occurs that can enable the radical transition to the more intuitively inspired "higher" levels.
Therefore in our culture, though the unconscious is still necessarily involved, its role is largely reduced to merely serving the needs of conscious type appreciation.
Now there were still many flaws with my approach at that time (especially in terms of adequately explaining the most advanced levels).
However the basic premise - and indeed all the basic insights - were well founded.
In other words the holistic mathematical approach serves as the appropriate basis for an integral scientific approach to development (which potentially entails the entire spectrum of stages).
Also this approach highlights the truly complementary nature of both physical and psychological processes (in horizontal, vertical and diagonal terms).
This entails therefore that just as we have many possible levels of psychological understanding, equally we have many possible levels of understanding with respect to the physical world.
So there is not just one valid scientific approach (that equates with Band 2 of the Spectrum)!
There are in fact many other more refined scientific approaches, that apply to the more advanced Bands. Therefore comprehensive understanding requires a truly multi-layered (i.e. multi-level) scientific approach.
So for simplification, I outline three major approaches to Mathematics and Science; 1) analytic 2) holistic and 3) radial, entailing the interpenetration of analytic and holistic.
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